The Inverse Scattering Transformation and The Theory of Solitons: An Introduction

THE INVERSE SCATTERING TRANSFORMATION AND THE THEORY OF SOLITONS

AN INTRODUCTION

f i KON.

I NED.

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NORTH-HOLLAND MATHEMATICS STUDIES 50

The Inverse Scattering Transformation and The Theory of Solitons

An Introduction

WKTOR ECKHAUS

AA#T VAN HARTEN

Mathernaticallnstitute State University Utrecht The Netherlands

NORTH-HOLLAND PUBLISHING COMPANY -AMSTERDAM NEW YORK 0 OXFORD

North-Holland Publishing Company, I981

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Eckhaue, Wiktor . of solitone.

The inverse scattering trausiormstion and the theory

(Nool-eh-Holland mathematics studies ; 50) Bibliography: p. Includes index. 1. 6cattering (Physics) 2. solitons. I. Barten,

A d i VUU, 1949- . 11. Tltle. 111. Series. QC20.7.83E25 530.1'5 81-1861 ISBN 0-444-86166-1 MCR2

PRINTED IN THE NETHERLANDS

PREFACE

The method of inverse scattering transformation and the theory

of solitons are among the most recent and fascinating achieve-

ments in the domain of applied mathematics. The development of

the theory, which took place mainly in the last decade, has

been explosive and far-reaching. It is not our ambition to

cover the whole field in this book. Our aim, in the first

place, is to provide an introduction for the uninitiated

reader, for the mathematician or physicist who has never heard

about solitons, or witnessed only from some distance the

excitement that they have created. Secondly however, we have

made it our goal to present the theory in a mathematically

satisfactory and rigorous way. As a result the book contains

various new points of view, and material that cannot be found

elsewhere. This, we hope, will also be of interest to

scientists who are experienced in the soliton theory.

The idea to write the book arose while teaching a course on the

inverse scattering transformation at the Mathematical Institute

in Utrecht, in the spring of 1979. We have experienced then

considerable difficulties in organizing and presenting the

material in such a way that it would be understandable and

acceptable to an uninitiated and rather critical audience.

The vast literature on the inverse scattering transformation,

integrable evolution equations and related topics, presents an

interwoven-pattern of various lines of development.

One of our difficulties was to find and follow a path suitable as an introduction into the field. Other difficulties came when

attempting to achieve a presentation that would be satisfactory

V

vi Preface

from the mathematical point of view. In the literature which is

strongly oriented towards physical applications, and which was

created during a rapid development, when one discovery was

leading to another, the reader is often confronted with

statements of which the demonstration is only outlined, or not

given at all. We have found that the task of filling in the

gaps, and straightening out the reasonings was not a trivial

one. Given all the labour thus invested in the course we have

decided to add some more, elaborate further the material, and

present this introduction to a larger audience.

The organization of the book follows essentially the historical

line of the development of the method of inverse scattering

transformation. This line is interruptedin chapters 4 and 5,

where an extensive and selfcontained treatment of the direct

and the inverse scattering problems for the Schrgdinger

equation and the generalized Zakharov-Shabat system is qiven.

In the last chapter (devoted to perturbations) we visit one of

the frontiers of the theory, where the analysis if still

largely in the heuristic stage, and where numerous questions

are open.

Many interesting topics are not touched upon in the book. We

list here, in an arbitrary order, some which we feel are

important. We do not discuss: the existence of conservation

laws, the Hamiltonian formalism, the periodic case for the

Korteweg-de Vries equation, discrete systems such as the Todda

chain, the Backlund transformation and the approach to inte-

grable evolution equations through the technique of exterior

differential forms. Again we stress that the book is mainly

meant as an introduction whichshould give the reader sufficient

knowledge to follow further his interests in the literature.

Survey papers, such as Ablowitz (1978), Miura (19761, Dubrovin,

Matveev & Novikov (1976), or the volume edited by Bullogh &

Caudrey (1980), will provide him with the bibliographical

leads.

Preface vi i

We gratefully acknowledge the contribution of Peter Schuur,

who read the manuscript, made us aware of various subtle

difficulties and helped us to solve them. The task of typing

has carefully and cheerfully been performed by Joke Stalpers

and Sineke Koorn.

Utrecht Wiktor Eckhaus

Aart van Harten


TABLE OF CONTENTS

PREFACE V

CHAPTER 1 : THE KORTEWEG-DE VRIES EQUATION

1.1. Historical introduction

1.2. Elementary properties

1.3. The soliton behaviour

1.4. The initial value problem. Existence and uniqueness of solutions

1.5. Miura's transformation and the modified K.d.V. equation 8

CHAPTER 2 : SOLUTION BY THE METHOD OF GARDNER-GREEN-KRUSKAL-

MIURA. THE INVERSE SCATTERING TRANSFORMATION

2.1. The scattering problem for the SchrBdinger equation on the line

2.2. Invariance of the spectrum for potentials satisfying the K.d.V. equation

2.3. Evolution of the scattering data

2.3.1. Evolution of the eigenfunctions

2.3.2. Evolution of the normalization

2.3.3. Evolution of the reflection

2.4. Summary and discussion of the method of solution by the inverse scattering transformation

coefficient Cn (t)

coefficient b (k, t)

2.5. The pure N-soliton solution

2.6. The pure 2-soliton solution: an exercise

2.7. Relation between soliton speed and eigenvalues

2.8. The emergence of solitons from arbitrary initial conditions

2.8.1. Formulation of the problem

13

1 4

16

20

20

21

24

26

29

35

38

,45

46

ix

X Table of Contents

2 . 8 . 2 . Analysis of Rc and Tc 4 7

2 . 8 . 3 . Solution of the Gel ' f and-Levitan equation 4 9

2 . 8 . 4 . Decomposition of the solution and estimates 5 0

CHAPTER 3 : ISOSPECTRAL POTENTIALS. THE LAX APPROACH 5 3

3 . 1 . The invariance of discrete eigenvalues by an elementary approach 5 8

3 . 2 . The invariance of the spectrum 6 0

3 . 3 . Isospectral potentials for the Schrodinger equation 6 4

3 . 4 . Isospectral potentials for more general selfadjoint operators 6 7

3 . 5 . An alternative approach 6 9

CHAPTER 4 : DIRECT AND INVERSE SCATTERING FOR THE

SCHRODINGER EQUATION 7 5

4 . 1 . Solutions and scattering data of Schrodinger's equation 7 8

4 . 2 . Properties of solutions 8 3

4 . 2 . 1 . Reformulation as integral equations 8 4

4 . 2 . 2 . Existence and uniqueness €or Im k > 0, k # 0 8 6

4 . 2 . 3 . Regularity for Im k 2 0, k Z 0 90

4 . 2 . 4 . Asymptotic behaviour

4 . 2 . 5 . The behaviour near k = 0

4 . 2 . 6 . Parmeter-dependent potentials

4 . 3 . The spectrum of - - dL + u on L 2 ( m ) dx2

4 . 4 . Fourier transform of the solutions

4 . 5 . Inverse scattering

4 . 6 . Concluding remarks

94

9 7

1 0 2

1 0 3

1 1 7

1 3 0

1 3 9

CHAPTER 5 : DIRECT AND INVERSE SCATTERING FOR THE

GENERALIZED ZAKHAROV-SHABAT SYSTEM 1 4 1

generalized Zakharov-Shabat system 1 4 3 5.1. Solutions and scattering coefficients of the

Table of Contents xi

5 . 2 . Properties of solutions 1 4 6 5 . 3 . The spectrum of (i -p) (&- 0) on L2 ( IRI2 155 5 . 4 . Fourier transform of solutions 158 5 . 5 . Inverse scattering 1 6 2

CHAPTER 6 : APPLICATIONS OF THE INVERSE SCATTERING

TRANSFORMATION 1 6 7 6 . 1 . The nonlinear Schrgdinger equation 1 6 8 6 . 2 . Isospectral potentials for nondegenerate first

order systems using an alternative approach 1 7 4 6 . 3 . Some evolution equations for isospectral

potentials by ad hoc procedures 1 7 7 6 . 4 . The general AKNS evolution equations 181 6 . 5 . Degenerate first order scattering systems and

the Sine-Gordon equation 1 8 7 6 . 6 . Higher order scattering systems 1 9 2

CHAPTER 7 : PERTURBATIONS 1 9 5 7 . 1 . Introduction and general formulation 1 9 5 7 . 2 . Evolution of the Scattering data in the case

of the Schr8dinger equation 1 9 8 7 . 2 . 1 . The discrete eigenvalues and the

reflection coefficient 1 9 8 7 . 2 . 2 . The normalization coefficient 200

7 . 3 . Evolution of the scattering data in the case of the generalized Zakharov-Shabat problem 20 3

7 . 3 . 1 . The discrete eigenvalues and the reflection coefficient 205

7 . 3 . 2 . The normalization coefficient 207 7 . 4 . Perturbation analysis 209

REFERENCES 217


CHAPTER 1

THE KORTEWEG-DE VRIES EQUATION

1.1. HISTORICAL INTRODUCTION

Most surveys and contemplative papers on the Korteweg-de Vries

equation start with a quotation from J. Scott-Russell's "Report

on Waves" (1844) describing his famous chase on horseback

behind a wave in a channel. Let us follow this tradition and

reproduce here Scott-Russell's glowing words:

"I was observing the motion of a boat which was rapidly drawn

along a narrow channel by a pair of horses, when the boat

suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of

,the vessel in a state of violent agitation, then suddenly

leaving it behind, rolled forward with great velocity, assuming

the form of a large solitary elevation, .a rounded, smooth and

well-defined heap of water, which continued its course along

the channel apparently without change of form or diminution of

speed. I followed it on horseback, and overtook it still

rolling on at a rate of some eight or nine miles an hour, pre-

serving its original figure some thirty feet long and a foot to

a foot and a half in hight. Its hight gradually diminished, and

after a chase of one or two miles I lost it in the windings of

the channel. Such, in the month of August 1834, was my first

chance interview with that singular and beautiful phenomenon..''

In the Korteweg-de Vries (1895) paper a mathematical model

equation was proposed, meant to provide, among other things, an

explanation of the phenomenon observed by Scott-Russell. In its

original form the equation reads as follows:

1

2 W. ECKHAUS & A. VAN HARTEN

where x is the variable along the (one dimensional channel), t

is time, q(x,t) is the elevation of the water surface above the

equilibrium level R, g the gravitation constant, ci a constant

related to the uniform motion of the liquid and u a constant

defined by

T is the surface copillarity tension and p the density.

The equation (1.1.1) is now known as the Korteweg-de Vries

equation, or KdV equation for short.

For some 65 long years the KdV equation led a quiet life,

mentioned occasionally in the literature, and occasionally

forgotten (Van der Blij (1978)). The break through came in

1960, when Gardner and Morikava rediscovered the equation as a

model for the analysis of collision-free hydromagnetic waves.

Since that date the Korteweg-de Vries equation has been re-

derived again and again, in different contexts, as a model

equation describing a considerable variety of physical pheno-

mena (see for example Miura (1976) and the literature quoted

there). To-day the Korteweg-de Vries equation can be considered

as one of the basic equations of mathematical physics. This

however is not its only claim to fame.

At least equally important is the development of new mathema-

tical methods and results, originating from the study of the

Korteweg-de Vries equation. This has led to applications

ranging from "practical" problems of wave propagation, to

rather ''pure" topics in algebraic geometry (see for example

Dubrovin, Matveev & Novikov (1976)).

It seems natural to ask the question, who were the men who

gave their names to the now famous equation, and in what way

did they collaborate? Some answers are given in Van der Blij

(1978) :

THE KDV EQUATION 3

Diederik Johannes Korteweg ( 3 1 . 3 . 1 8 4 8 - 5 . 1 0 . 1 9 4 1 ) was a well-

known Dutch mathematician professor at the University of

Amsterdam and author of numerous papers. It is curious to note

that in several necrologies published after his death, the

Korteweg-de Vries ( 1 8 9 5 ) paper appears not be mentioned.

Gustav de Vries wrote a Doctoral Desertation under Korteweg's

supervision, and presented it at the university of Amsterdam

on December 1 , 1 8 9 4 . The thesis was in Dutch, and featured

on page 9 the equation now known as the K.d.V. equation. It

seems that G. de Vries has spent most of his further professio-

nal life as a high-school teacher.

Let us finally remark that the K.d.V. equation, in spite of its

fame and popularity, has not remained unchallenged as a model

equation describing the behaviour of (long) water waves in a

channel. Recently Benjamin, Bona & Mahony ( 1 9 7 2 ) have proposed

an alternative model. A discussion of these matters can be

found for example in Kruskal ( 1 9 7 5 ) .

1 . 2 . ELEMENTARY PROPERTIES

One obtains standard forms of the Korteweg-de Vries equation by

transformation of variables which removes from the equation all

reference to the original physical problem. A form that is much

used arises through the transformation

one then obtains -.

The numerical factor in front of the second term does not have

any particular significance. In fact, by modifying the trans-

formations x,t -+ ;,:, I-I + u one can obtain

au au au a% ..- + p= + v u y + y-q = 0 a; ax ax ax

( 1 . 2 . 3 )

where v , v, Y, v # 0, y # 0, are numerical factors that can be

chosen at will. However, we shall adhere here to the widespread


preference to the form (1.2.2). Dropping the bars over the

variables, the Korteweg-de Vries equation will be €or us:

= o (1.2.4)

We remark that the equation has the property of Galilean in-

variance, in the following sense:

ut - 6uuX + u xxx

Consider the transformation

(1.2.5)

then u* satisfies

t* = t ; x* = x-ct ; u*(x*,t*) = u(x*+ct*,t*) + zc 1

u:*- ~u*u:, + u** ,, , = 0 x x x (1.2.6)

We consider briefly the l i n e a r i z e d K . d . V . e q u a t i o n , i.e.

(1.2.7) Ut + uxxx = 0

The equation admits as solutions harmonic waves

(1.2.8) u(x,t) = Ae

provided that, for each wavenumber k , the phase speed c satisfies

(1.2.9) c = - k .

Waves for which the phase speed is not constant (as a function

of the wavenumber) are called d i s p e r s i v e (Whitham (1974)). The

relation (1.2.9) is called the dispersion relation. Since the

equation (1.2.7) is linear, any superposition of harmonic

waves (with different wavenumbers) I s again a solution of

(1.2.7). We note that all dispersive wave solutions of the

linearized K.d.V. equation t r a v e l t o t h e l e f t (with increasing

time).

ik (x-ct)

2

We now return to the full K.d.V. equation and look for the

existence of special solutions called waves of permanent type

(Lamb (1932)), also called travelling waves or progressing

waves. These are waves which, when viewed in some particular

moving coordinate system, have a shape that does not change

with time. We thus pose:

(1.2.10) u(x,t) = U(x-ct)

THE KDV EQUATION 5

Substitution into the K.d.V. equation leads, for the function

U(z) , to the ordinary non-linear differential equation

(1.2.11) U"'- (6Utc)U' = 0

where the primes denote differentiation. Integrating once we

get:

(1.2.12) U" - 3u 2 - c ~ = m

where m is an arbitrary constant. Multiplying bu u' and inte-

grating again we find

(1.2.13)

where n again is an arbitrary constant.

UI2 - 2U3 - cU2 - 2mU = n

In the final stage U can implicitly be defined in terms of

elliptic integrals. From that result one can derive the

existence of periodic solutions U(z) = U(z+T), which can be

expressed in terms of the Jacobi elliptic functions cn, and

are therefore called c n o i d a l w a v e s (see for example Whitham

(1974) for details).

In all that follows, of particular importance will be solutions

of permanent type U(z) which are such that U and its deriva-

tives vanish for z + 7". These solutions will be called

s o l i t a r y w a v e s .

For a solitary wave we can put in (1.2.12) m=O and in (1.2.13)

n=O. We thus have

(1.2.14) Ul2 = u (2u+c) 2

The equation can simply be integrated, and one finds

(1.2.15) u(x,t) = U(x-ct) = -2c 1 sech2[$fi(x-ct+xo) I

where xo is an arbitrary constant. Furthermore:

(1.2.16) sech z =

We thus see that the solitare wave decays exponentially for

4 - - 2 1 2 -2 2 (cosh 2) (e +e 1

2 + i m .

Two observations are further of importance:


The solitary wave solution exists only for c > 0. Hence, any solitary wave of the K.d.V. equation moves t o t h e r i g h t (with

increasing t) . The propagation speed of the solitary wave c is proportional to

the amplitude of the wave (which equals -#c). Hence, a Zarger

s o Z i t a r y wave moves f a s t e r t h a n a smaZZer o n e .

1.3. THE SOLITON BEHAVIOUR

Because the K.d.V. equation is nonlinear, any superposition of

solitary wave solutions will not be a solution of the equation.

This observation may Lead one to think that the importance of

the solitary waves in the general theory of the K.d.V.

equation, will be a very limited one. A first indication in the

opposite direction came from the work of Zabusky E Kruskal

(1965) and Zabusky (1967).

Let us denote the function describing a solitary wave by

(1.3.1)

and let us imagine the following experiment:

S(z,c) = -ic sech 2 1 [~fiz]

At t=O the value u(x,O) of a solution u(x,t) of the K.d.V.

equation is given by

(1.3.2)

where X > 0 and sufficiently large, and c solitary waves decay exponentially, at the initial time the

two solitary waves do not interfere much. Since c1 > c2 one should expect that the larger solitary wave will tend to catch

up with the smaller one. What will be the effect of the inter-

action?

U(X,O) = S(X,Cl) + S(x-X,c2) > c2. Because the 1

In the work of Zabusky and Kruskal, the experiment has been

performed by numerical analysis, and led to the following

result :

For t = T > 0, sufficiently large, one has (1.3.3) u(x,T) = S(x-clT-Bl,cl) + S(x-c2T-e2,C2)

THE KDV EQUATION 7

where e l and e 2 are constants.

Thus, the two solitary waves emerge after interaction as two

solitary waves unchanged in shape. The only effect of the

interaction is represented by phaseshifts

the two solitary waves retain their entities through inter-

action, Kruskal and Zabusky coined €or them the name s o l i t o n s .

suggesting a particle-like behaviour. The term soliton has

become immensely popular, in particular in mathematical physics.

There does not seem to exist a mathematical definition of what

a soliton is; usually the definition is given in the context of

some particular problem, through a formula (see for example

Miura (1976) section 6).

and 0 2 . Because

1.4. THE INITIAL VALUE PROBLEM. EXISTENCE AND UNIQUENESS OF SOLUTIONS

Let u(x,t) be defined as a solution of

(1.4.1) ut - 6uuX t uxx = 0, x E(-m,m), t > 0 U(X,O) = uo(x)

Bona and Smith (1975) have demonstrated the existence of a

classical solution if uo(x) and its derivatives up to fourth

one are squared-integrable. Further results on the existence

and regularity of solutions are given in Tanaka (1974) and

Cohen (1979). It appears that there is a strong relation

between the regularity properties of u(x,t), t > 0, and the way uo(x) and its derivatives decay for 1x1 + 00. The faster

u (x) and its derivatives decay, the smoother the solution u(x,t), t > 0 will be. From Cohen (1979) it follows that if uo(x) and its first four derivatives decay faster than

Ix~'~, Vn, as 1x1 + m, then the solution u(x,t), t > 0 will be infinitely differentiable.

0

Uniqueness of solutions within a class of functions which,

together with a sufficient number of derivatives, vanish for

1x1 + 0 3 , can easily be demonstrated following Lax (1968).

We reproduce the proof here.


Let u and ( 1 . 4 . 1 ) and consider

( 1 . 4 . 2 ) w = u - u

be two solutions of the initial value problem

z

Then - - =

xxx aw 6uux - 6uux - w at ( 1 . 4 . 3 )

After some trivial manipulations one obtains €or w the linear

equation 5 aw - = 6uwX + 6uXw - w at xxx ( 1 . 4 . 4 )

We multiply by w and integrate: m m m m

2 dt -00 -m -m -m

2 - 2 wxxxdX

( 1 . 4 . 5 ) - - I w dx = 6 I uwwxdx + 6 I uxw dx -

If w, wx and wxx tend to zero as 1x1 --* m, then the last term

on the right hand side of ( 1 . 4 . 5 ) can easily be shown to be

equal to zero. Furthermore, integrating by parts in the first

term of the right hand side of ( 1 . 4 . 5 ) one finds

2 - m m d 2 ( 1 . 4 . 6 ) - dt I w dx = 1 2 (ux-+ux)w dx

-m -m

We now use 5

( 1 . 4 . 7 ) Iux-%UxI M , x E(-m,m)

to obtain

( 1 . 4 . 8 ) $ w dx 12M w dx - m 2 m 2

dt -m -m

From this differential inequality it follows that

m 2 1 2 M t m

2 ( 1 . 4 . 9 ) I w dx Q [ I w XI^,^ e -m -m

However, for t=O, w as defined in ( 1 . 4 . 2 ) equals zero, because

u and both satisfy the same initial condition of the problem

( 1 . 4 . 1 ) . Hence w=O for t > 0, which proves the uniqueness of solutions of ( 1 . 4 . 1 ) .

1 . 5 . MIURA'S TRANSFORMATION AND THE MODIFIED K.D.V. EQUATION

In the mathematical literature there exist examples of trans-

THE KDV EQUATION 9

formations by which solutions of some linear differential

equation generate solutions of an associated nonlinear

equation. A rather elementary example is given by:

Lemma 1.5.1. L e t v(x) b e a s o l u t i o n of t h e S c h r d d i n g e r e q u a t i o n

v - u(x)v = 0 xx t h e n t h e f u n c t i o n w(x), d e f i n e d b y

"X w = - V

i s a s o l u t i o n of t h e R i c c a t i e q u a t i o n

2 wx + w = u.

Proof of the lemma is obtained by straightforward substitution.

We mention further a slight but important generalization, i.e.:

If v(x) satisfies

(1.5.1) v - (u(x)-X)v = 0

with X an arbitrary constant, then

(1.5.2)

xx

vX w = - X

satisfies

(1.5.3) + w2 = u - A . wX

A more sophisticated result, analogous to Lemma 1.5.1, has been

discovered by Hopf (1950) and Cole (1951).

Lemma 1.5.2. L e t v(x,t) b e a s o l u t i o n o f t h e h e a t - e q u a t i o n

- Vt - vvxx

t h e n t h e f u n c t i o n w(x,t), d e f i n e d b y t h e Hopf-Cole t r a n s -

f o r m a t i o n V

V X w = -2v -

s a t i s f i e s t h e Burgers e q u a t i o n

Wt + wwx - - vwxx

The proof follows again by substitution.


Note that the Burgers equation resembles somewhat the K.d.V.

equation. In view of the existence of the results of the type

stated in Lemma 1.5.2 it seems natural to search for an ana-

logous transformation for the Korteweg-de Vries equation.

The following result is due to Miura (1968):

Lemma 1.5.3. L e t w(x,t) b e a s o Z u t i o n o f t h e m o d i f i e d K.d ." .

e q u a t i o n

= o w t - 6 w w 2 + W X xxx

t h e n t h e f u n c t i o n u(x,t) d e f i n e d by t h e Miura t r a n s f o r m a t i o n .. L

+ wx u = w

s a t i s f i e s t h e K . d . V . e q u a t i o n .

ut - 6uuX + uxXx = 0 .

Again, the proof is by substitution.

We note that, as compared to the results given in Lemma 1.5.1

and 1.5.2, Miura's transformation works in a "wrong direction":

solutions of the nonlinear K.d.V. equation are generated by

solutions of an equation with a stronger nonlinearity.

Suppose now that one attempts to interprete Miura's trans-

formation in the inverse direction, as a transformation which

defines a function w in terms of the function u. Then w is a solution of Riccati's equation! One may generalize somewhat

further, because of the Galilean invariance of the K.d.V.

equation, and write

2 (1.5.4) u - x = w + w x

then, through egs. (1.5.1) , (1.5.21, (1.5.3) one is led to consider as associated to the K.d.V. equation, the Schr6dinger

equation (1.5.1), with a potential u that satisfies the K.d.V.

equation.

The reader may find the considerations given above, following

Lemma 1.5.3, little convincing and not very deductive. Never-

theless, it is a reasoning of this type that is often used

THE KDV EQUATION 11

(Kruskal (1975), Miura (1976)) to motivate the first and

essential step in the surprising discovery by Gardner, Greene,

Kruskal and Miura (1967) of a method of solution of the

initial value problem for the Korteweg de Vries equation.


CHAPTER 2

SOLUTION BY THE METHOD OF GARDNER-GREENE-KRUSKAL-MIURA THE INVERSE SCATTERING TRANSFORMATION

I n a series of s u r p r i s i n g and remarkable d i s c o v e r i e s Gardner , Greene, Kruskal and Miura (GGKM f o r s h o r t ) have developed a method o f s o l u t i o n f o r t h e K.d.v. e q u a t i o n which l a t e r , through v a r i o u s g e n e r a l i z a t i o n s , h a s become known as t h e method o f i n v e r s e s c a t t e r i n g t r a n s f o r m a t i o n (also c a l l e d s p e c t r a l t r a n s - format ion o r i n v e r s e s c a t t e r i n g t r a n s f o r m ) . W e s h a l l d e s c r i b e t h e f u r t h e r developments i n c h a p t e r s 3 and 6 , where it w i l l a l s o appear t h a t v a r i o u s s t e p s of t h e o r i g i n a l GGKM a n a l y s i s can be s i m p l i f i e d . I t o f t e n o c c u r s i n mathematics t h a t , once a r e s u l t h a s been e s t a b l i s h e d , a new and s i m p l e r demons t r a t ion can be g iven . However, t h e d i s c o v e r y c a n b e t t e r b e a p p r e c i a t e d b y fo l lowing t h e o r i g i n a l r eason ing . Fur thermore t h e GGKM a n a l y s i s remains of i n t e r e s t because of i t s i n g e n u i t y .

The main p a r t o f t h i s c h a p t e r i s devoted t o t h e d e s c r i p t i o n o f t h e GGKM-method and r e s u l t s ( s e c t i o n s 2 . 1 t o 2 . 6 ) . I n t h i s w e fo l lows t h e o r i g i n a l pape r s G G K M ( 1 9 6 7 ) , (1974) w i t h o n l y some minor m o d i f i c a t i o n s , and some a d d i t i o n a l c o n s i d e r a t i o n s which s e r v e to t i g h t e n up t h e ma themat i ca l r eason ing . I n s e c t i o n 2 . 7 an impor t an t r e s u l t due t o Lax(1967) i s d e r i v e d . The f i n a l s e c t i o n d e s c r i b e s r e c e n t r e s u l t s on t h e behaviour f o r l a r g e t i m e o f s o l u t i o n s o f t h e K.d.v. e q u a t i o n w i t h a r b i t r a r y i n i t i a l c o n d i t i o n s .

The s t a r t i n g p o i n t o f t h e GGKM method i s t h e i n t r o d u c t i o n o f t h e f u n c t i o n s u ( x , t ) t h a t s a t i s f y t h e Korteweg-de V r i e s e q u a t i o n

13

1 4 W. ECKHAUS & A. VAN HARTEN

as p o t e n t i a l s i n t h e Schr6dinger e q u a t i o n

( 2 . 2 ) v X x - { u ( x , ~ ) - A } v = 0, x E (-a,-).

2 . 1 . THE SCATTERING PROBLEM FOR THE SCHRODINGER EQUATION ON

THE L I N E .

T h i s s e c t i o n summarizes t h e main r e s u l t s o f t h e a n a l y s i s which w i l l be d e s c r i b e d i n d e t a i l i n Chapter 4 . The r e s u l t s w i l l s e r v e as t o o l s o f a n a l y s i s i n t h e p r e s e n t c h a p t e r . For n o t a t i o n a l s i m p l i c i t y w e s u p p r e s s i n t h i s s e c t i o n t h e t i m e dependence o f t h e p o t e n t i a l and c o n s i d e r

v - xx ( 2 . 1 . 1 )

W e assume t h a t t h e

( 2 . 1 . 2 ) 7 l u ( x -m

p o t e n t i a l s a t i s f i e s t h e c o n d i t i o n

11x1 dx < m, k = 0 , 1 , 2 . k

We s e a r c h f o r v a l u e s o f 1 ( c a l l e d t h e e i g e n v a l u e s ) f o r which t h e r e e x i s t s o l u t i o n s v ( x ) o f t h e e q u a t i o n ( 2 . 1 . 1 ) which are bounded as 1x1 + m . The c o l l e c t i o n o f a l l e i g e n v a l u e s w i l l be c a l l e d t h e spec t rum cor re spond ing t o a g iven p o t e n t i a l u ( x ) . ( A more c a r e f u l 1 d e f i n i t i o n of t h e spec t rum is g i v e n i n Chapter 3 and 4 ) . From c h a p t e r 4 w e have t h e f o l l o w i n g r e s u l t s :

For each p o t e n t i a l s a t i s f y i n g ( 2 . 1 . 2 ) t h e r e e x i s t s a f i n i t e number ( p o s s i b l y zero) of d i s c r e t e s imple e i g e n v a l u e s

(2 .1 .3) 2 x = x = -kn, n kn IR+

which are such t h a t t h e co r re spond ing e i g e n f u n c t i o n s $,(x) be long t o L2 ( IR). We s h a l l t a k e t h e e i g e n f u n c t i o n s t o be normal ized by:

( 2 . 1 . 4 ) m

I $:(x)dx = 1, J ln(x) > 0 f o r x + + m

-m

INVERSE SCATTERING TRANSFORMATION 15

The behaviour of t h e s e e i q e n f u n c t i o n s , f o r x + f. m, i s g iven by :

-knx Gn ( X I - C nen €or x + m

One can thus d e f i n e t h e no rma l i za t ion c o e f f i c i e n t s

There a l s o e x i s t s o l u t i o n s o f t h e Schrddinqer e q u a t i o n which a r e bounded f o r 1x1 + m, f o r

2 ( 2 . 1 . 7 ) A = + k , V k E I R , k # O .

These s o l u t i o n s t o be i n d i c a t e d by $,(x) behave f o r x + T m as a l i n e a r combinat ion of

+ikx and e - ikx e

W e d e f i n e s o l u t i o n s $k ( x ) th rough t h e fo l lowing n o r m a l i z a t i o n

e-kx+b (k) eikx

a ( k ) e-ikx

f o r x + +m

f o r x + -a

(2 .1 .8) $,(x) -

a ( k ) i s c a l l e d t h e t r ansmiss ion c o e f f i c i e n t and b ( k ) t h e r e f l e c t i o n c o e f f i c i e n t . They are r e l a t e d by t h e c o n s e r v a t i o n law:

( 2 . 1 . 9 ) + lb12 = 1

One can normal ize t h e €unc t ions qn and qk i n a way d i f f e r e n t from t h e one in t roduced above. This w i l l be e x p l a i n e d i n Chapter 4 , where t h e r e l a t i o n s between d i f f e r e n t n o r m a l i z a t i o n s w i l l a lso be s t u d i e d . I n g e n e r a l t h e use of a p a r t i c u l a r no rma l i za t ion i s mainly a matter of tas te , and sometimes a matter o f convenience.

The spectrum o f t h e SchrEdinger e q u a t i o n , t o g e t h e r w i t h t h e


c o e f f i c i e n t s Cn ,a (k ) , b ( k ) are c a l l e d the s c a t t e r i n g d a t a o f a g iven p o t e n t i a l u ( x ) . W e now t u r n t o t h e i n v e r s e s c a t t e r i n g problem, which c o n s i s t s of de t e rmin ing t h e p o t e n t i a l u ( x ) from i t s s c a t t e r i n g d a t a . The a n a l y s i s o f c h a p t e r 4 p r o v i d e s t h e s o l u t i o n as f o l l o w s :

W e d e f i n e a f u n c t i o n B ( 5 ) t h rough :

m 2 -kn' + 1 J b (k)eikr;dk. ( 2 . 1 . l o ) B ( 5 ) = z C n e

n= 1 -m

Where N i s t h e number of d i s c r e t e e i g e n v a l u e s . The f i r s t t e r m on t h e r i g h t hand side o f (2 .2 .10) i s a b s e n t i f t h e r e are no d i s c r e t e e i g e n v a l u e s . W e f u r t h e r d e f i n e t h e f u n c t i o n K (x ,y ) a s t h e s o l u t i o n of t h e i n t e g r a l e q u a t i o n

m

( 2 . l . 11) K(x ,y) + B(x+y) + B ( z + y ) K ( x , z ) d z = 0 , y > x. X

Then :

( 2 . 1 . 1 2 )

The i n t e g r a l e q u a t i o n ( 2 . 2 . 1 1 ) i s u s u a l l y c a l l e d t h e Ge l ' f and-Lev i t an e q u a t i o n , w h i l e some a u t h o r s p r e f e r t o c a l l it t h e Marchenko-equation. Re fe rences t o t h e l i t e r a t u r e on t h e i n v e r s e s c a t t e r i n q t h e o r y w i l l b e g i v e n i n c h a p t e r 4 .

2.2 . INVARIANCE OF THE SPECTRUM FOR POTENTIALS SATISFYING THE K . D . V . EQUATION.

L e t now u ( x , t ) be a f u n c t i o n t h a t s a t i s f i e s

ut - 6uuX + u = 0 , x E ( - m , m ) , t > 0 xxx ( 2 . 2 . 1 )

U ( X r 0 ) = u O ( X I

and c o n s i d e r t h e one pa rame te r f a m i l y of S c h r 6 d i n g e r e q u a t i o n s .


The s c a t t e r i n g d a t a can be computed for t = 0 , because u(x ,O) is a g iven f u n c t i o n . We s h a l l s t u d y t h e e v o l u t i o n o f t h e s c a t t e r i n g d a t a f o r t > 0 u s i n q as t h e o n l y in fo rma t ion t h e f a c t t h a t u ( x , t ) s a t i s f i e s t h e K.d.V. e q u a t i o n .

The fundamental r e s u l t concern ing t h e spec t rum i s as f o l l o w s :

Theorem 2 . 2 . 1 . L e t u ( x , t ) b e a s o l u t i o n of t h e Korteweg-de

V r i e s e q u a t i o n wh ich s a t i s f i e s c o n d i t i o n ( 2 . 1 . 2 ) and w h i c h i s

s u c h t h a t for p = 1,2,3

is bounded f o r 1x1 * 00. Then t h e c o r r e s p o n d i n g s p e c t r u m o f t h e

S c h r d d i n g e r e q u a t i o n is i n v a r i a n t i n t i m e .

The proof of t h e a s s e r t i o n o f t h e theorem i s t r i v i a l f o r t h e cont inuous p a r t o f t h e spectrum h = k 2 , g iven t h e f a c t t h a t u ( x , t ) s a t i s f i e s t h e c o n d i t i o n ( 2 . 1 . 2 ) . We t h e r e f o r e t u r n t o t h e d i s c r e t e p a r t o f t h e spec t rum.

L e t X = -kn be an i s o l a t e d d i s c r e t e s imple e i g e n v a l u e €or t = O . Because t h e p o t e n t i a l u ( x , t ) i s a con t inuous ly d i f f e r e n t i a b l e f u n c t i o n o f t h e parameter t , one can deduce t h a t t h e r e e x i s t s a cont inuous fami ly o f d i s c r e t e e i g e n v a l u e s X = l ( t ) w i t h X ( 0 ) = -kn. Furthermore, A ( t ) i s d i f f e r e n t i a b l e (see Ch. 4 ) .

2

2

L e t t h e co r re spond ing fami ly of e i g e n f u n c t i o n s , normal ized by ( 2 . 1 . 4 ) , be denoted by $ ( x , t ) . W e have :

(2 .2 .3)

I n c h a p t e r 4 , i t is shown t h a t $ ( x , t ) i s con t inuous ly d i f f e r e n t i a b l e w i t h r e s p e c t t o t.

JIxx - I u ( x , t ) - X ( t ) I $ ='O

With t h e s e p r e l i m i n a r i e s w e can fo rmula t e a r e s u l t which p l a y s an impor t an t role i n t h e a n a l y s i s of GGKM:

Lemma 2 . 2 . 1 . L e t t h e p o t e n t i a Z u ( x , t ) s a t i s f y t h e K . d . V .

e q u a t i o n and l e t A = X ( t ) b e a f a m i Z y of i s o Z a t e d e i g e n v a l u e s

18 W. ECKHAUS & A . VAN HARTEN

w i t h corresponding eigenfunctions $(x,t). One then has the

following reZation

w i t h

Proof of the lemma proceeds essentially by substitution and

manipulation of formulas. We outline the main steps.

Differentiating the SchrGdinger equation with respects to t one

gets

a2 (2.2.4) 1 7 - (u-A)l$t = (ut-At)$. ax

a2 ax

Using the K.d.V. equation to get rid of ut we find

(2.2.5) I $ + At$ = 0. [T - (u-A) I $t - (6uux-uxXx

We now write

(2.2.6)

and using the SchrGdinger equation obtain

a' u xxx JI = ---& UX$ - UXQXX - 2UXX$,

a L ax

u J, = [T - (u-A)l u X - 2UXX$,. xxx (2.2.7)

Thus we have, at this stage

a L ax

(2.2.8) [T - (u-A)] (J,t+~x$) - ~ ( ~ u u ~ ' ~ . J + u ~ ~ $ ~ ) + At$ = O

The final step is to show that

a 2 3uux$ + UXX$, = 1 7 - (u-X)I (U+2A)JlX. ax

( 2 . 2 . 9 )

This is left as an excercise to the reader.

We now proceed to the proof of the theorem 2.2.1. Multiplying

the re1at.m given in lemma 2.2.1 by J, we have

INVERSE

2 (2.2.10) -At+ = +

SCATTERING TRANSFORMATION

2 M - (u-X)$M

ax

19

Then, using the Schrbdinger equation, we obtain:

Finally, integrating over x, yields:

(2.2.12)

We now consider the behaviour of the derivatives of +(x,t) for

1x1 + m . Because the function $(x,t) tends to zero (exponent-

ially) for 1x1 + m, from the Schrgdinger equation (with u(x,t)

being bouded) we have that the same is true for the function

Jlx,(x,t). One now easily deduces, by elementary interpolation

between seminorms , that the function $,(x,t) also tends to

zero for 1x1 + m .

m

-At = [$Mx - qxM1 / -m

It is slightly more difficult to investigate the behaviour of

the function +,(x,t) as 1x1 + 03. Reasoning from the equation

(2.2.5), which +t satisfies, one can deduce that if 0, would contain terms which do not tend to zero, then such terms would

grow exponentially as 1x1 + m . The presence of such terms in

the function $,(x,t) would then be in contradiction with the

behaviour of $(x,t) as 1x1 + m . The conclusion is that $t(x,t)

also tends to zero for 1x1 + m .

One now easily.verifies that a l l terms on the right hand side of the equation (2.2.13) tend to zero for x + fm.

This proves that any discrete eigenvalue X = -kL for t=O

remains an eigenvalue for all t > 0 for which u(x,t) satisfies the conditions specified in Theorem 2.2.1. We must finally

show that no new eigenvalues can be created at some t = to > 0. Suppose the contrary, i.e. that there exists for t = to an

eigenvalue X = -u be a continuous family X = h(t) with X(tO) = -u , h(t) a differentiable function and Xt = 0, by the preceeding analysis.

However, from the theory of Chapter 4 we know that if an eigen-

n

2 # -kn, n = 1, . . .N. Then again there would 2


value is created, then it must start out from the origin, and

this contradicts the results of the reasoning given above.

2.3. EVOLUTION OF THE SCATTERING DATA.

2.3.1. Evolution of the eigenfunctions.

The lemma 2.2.1 is not only useful in the proof of theorem

2.2.1 but also leads to further important results in the study

of the evolution of the scattering data. We observe that the

lemma also holds when we consider the continuous part of the

spectrum, i.e. for X = k2, with Xt equal to zero. This assertion is easily verified by glancing over the proof of the Lemma.

Combining the results we have:

Lemma 2.3.1.1. Let the potentiaZ u(x,t) be as specified in

Theorem 2 . 2 . 1 . Let h be any point of the spectrum and $(x,t)

the corresponding eigenfunction. T h e n t h e function M defined by

M = Jlt - 2(~+2X)$, + uX$

satisfies the Schradinger equation

Solving the Schrcdinger equation for M we find

where 0, for each A , is a solution of the Schrodinger equation that is linearly independent with the eigenfunction $. C and

D are, at this stage, arbitrary constants.

We shall now show that D equals zero. For the discrete spec-

trum X = -k2 the reasoning is elementary: any function Q, that

satisfies the Schrbdinger equation and is linearly independent

with the eigenfunctioe $n will contain terms which behave as

e nx for x + m and e

left hand side of the equation (2.3.1.1) tend to zero as

1x1 + m. Hence D must be zero.

n

- nx for x -+ -a l while all terms on the


For the continuous part of the spectrum X = k2 the reasoning

is somewhat more involved. We consider the equation (2.3.1.1)

for x --* -00. Then +(x,t) behaves as e -ikxl and from the

analysis of chapter 4 it follows that the same is true for the

function +,(x,t). The behaviour of the function +t(x,t) can be

investigated in a way analogous to what has been described in

section 2.2 for the eigenfunctions corresponding to the dis-

crete spectrum. One then finds that + (x,t) also behaves as -ik x e for x + -00.

t

NOW, a solution of the Schrgdinger equation for X = kL that is

linearly independent to the eigenfunction defined in equation

(2.1.8) will have the following behaviour for 1x1 -+ 00

g(k)e+ikx for x -+ +a r

for x -+ -m

Comparing the behaviour ofthe right hand side and the left

hand side of the equation (2.3.1.1) for x + -00 we deduce that

D = 0.

We have hence found that if X is any point of the spectrum,

the corresponding eigenfunction satisfies

The above equation can be considered as an evolution equation

for the eigenfunctions. The constant C is as yet undetermined.

We shall find that C takes different values for the continuous

and the discrete parts of the spectrum.

In the subsections that follow now we assume that u(x,t) and

ux(x,t) tend to zero as 1x1 + 00, uniformly with respect to t

on any compact time interval.

2.3.2. Evolution of the normalization coefficient Cn(t).

2 Let X = -kn be a point of the discrete spectrum. The eigen-

functions are normalized by


O 2 (2.3.2.1) $ dx = 1

This requirement will determine the coefficien- C in he

evolution equation for eigenfunctions (2.3.1.3). We multiply

that equation by the function $ and integrate over x. We then

have :

-m

m m

2 dt -m -m -m

I m 2 $ dx = 1[2(~+2X)$$~-~l~$ 2 ]dx + C I $ 2 dx (2.3.2.2) - -

It is an amusing exercise, which we leave to the reader, to

show that

(2.3.2.3) S[~(U+~X)$$~-U~$ Idx = 0

Using eq. (2.3.2.1) it follows that

(2.3.2.4) C = 0.

The evolution equation for an eigenfunctionwhich corresponds to

a discrete eigenvalue h = -kn is thus fully determined, and

reads :

(2.3.2.5)

m 2

-m

2

2 $t = 2 ( ~ - 2 k ~ ) $ ~ - ux$

In order to study the behaviour for x + m we introduce the

function w(x,t) by

(2.3.2.6) w(x,t) = e $(x,t)

We are given that

(2.3.2.7) lim w(x,t) = Cn(t)

Furthermore, from the analysis of chapter 4 it follows that

knx

X+m

(2.3.2.8) lim wx(x,t) = 0 X+W

Both limits (2.3.2.7) and (2.3.2.8) are uniform with respect

to t in any compact interval, for which the potential u(x,t)

satisfies the basic condition (2.1.2).

Introducing w(x,t) by eq. (2.3.2.6) into the evolution


equation (2.3.2.4) we obtain

(2.3.2.9) wt = 4knw 3 + 2(u-2kn)wx 2 - (2knu+ux)w

Next, by an elementary "variation of a constant" formalism, we

deduce

4kit (2.3.2.10) w(x,t) = w(x,O)e +

3 t 4kn (t-t I ) + S e 0

2 {~[u(x, t' ) -2kn]wx (x, t') -

The expression between brockets under the integral sign tends

to zero uniformly with respect to t as x + m. Therefore, taking

the limit in eg. (2.3.2.10) we may interchange integration and

the limit process, and obtain

(2.3.2.11) lim w(x,t) = Cn(t) = lim w(x,O)e = Cn(0)e 4kit 4k:t

X-tm X+m

We have thus established

Theorem 2.3.1. L e t t h e p o t e n t i a l u(x,t) s a t i s f y t h e c o n d i t i o n s

of t heorem 2 . 2 . I , and f u r t h e r m o r e

lim u(x,t) = lim ux(x,t) = 0 1x1- 1x1-

u n i f o r m l y w i t h r e s p e c t t o t on any compact t i m e i n t e r v a l .

L e t A = -kn b y any d i s c r e t e e i g e n v a l u e , w i t h a c o r r e s p o n d i n g 2

e i g e n f u n c t i o n J I (x, t) n o r m a l i z e d b y m

2 J J I (x,t)dx = 1 -m

Then t h e n o r m a l i z a t i o n c o e f f i c i e n t

Cn(t) = lim e JI(x,t) knx

X'm

Cn(t) d e f i n e d by

4kit i s g i v e n by Cn(t) = Cn(0)e .


2.3.3. Evolution of the reflection coefficient b(k,t).

We now consider X = k2. The evolution of the corresponding

eigenfunction 9 (x,t) is described by

The analysis of the behaviour for x -+ m is somewhat more in-

volved than in the preceding section. We shall therefore first

derive the correct result by a simple, heuristic, but rather

non-rigorous argument. We shall then proceed to justify the

result.

For x + we approximate eq. (2.3.3.1) by

(2.3.3.2) 2 JIt % 4k (Jx + C(J

Next we substitute, using the asymptotic behaviour of Q(x,t)

(2.3.3.3) Qt bte ik x

+ beikX I a -ikx JIx = z [ e

This produces:

(2.3.3.4)

The first conclusion is:

(2.3.3.5) C = 4ik

We are then left with

(2.3.3.6) bt = 8ik3b

which yields

(2.3.3.7) b(k,t) = b(k,O)e

3 -ik x bteikX = (4ik3+C)beikx + (-4ik +C)e

3

8ik3t

In order to reproduce this result by a rigorous analysis we

write

w(l) (x,t) + e ikx w(2) (x,t) -ik x (2.3.3.8) $(x,t) = e

We are given that


Furthermore, from the analysis of chapter 4 it follows that

(2.3.3.10) lim wL1) (x,t) = lim w ( ~ ) (x,t) = 0 X+m X+m

All limits in (2.3.3.9), (2.3.3.10) are uniform with respect

to t in compact intervals for which the potential u(x,t)

satisfies the condition (2.1.2) . We introduce the decomposition (2.3.3.8) into the evolution

equation (2.3.3.1), rearrange somewhat, multiply through by

the factor eikx and integrate over time. These operations

produce the following result:

t

0 (2.3.3.11) w ( l ) (x,t) - w(') (x,O) + (4ik3-C) J w(')dt' +

t

0 + J [-2 (u+2k2)wA1) + (2iku+u,)w(') ldt' =

t = - e 2ikx (w(~) (x,t)-w(2) (x,O)-(lik 3 +C) w(2)dt' t

0 t

0 + J [-2 (u+2k2)wi2) + (-2iku+~~)w(~)]dt'}.

We observe that for the expression on the left hand side the

limit for x + 00 exists. For the expression on the right hand

side the limit does not exist, unless the limit of the ex-

pression between brackets, which does exist, is zero.

Taking the limit, by an argument similar to the preceeding

section, we obtain

(2.3.3.11) t

lim {w") (x,t)-w(') (x,0)+(4ik -C) .f w("dt'1 = 0 X+= 0

3

t 3

(2.3.3.12) lim {w(~) (x,t)-w(2) (x,0)-(4ik +C) I w(2)dt') = 0 . x+m 0

From eq. (2.3.3.11),using the first part of (2.3.3.91, it

follows that


(2.3.3.13)

From eq. (2.3.3.12), using the second part of (2.3.3.91, we

have

(2.3.3.14) b(k,t) = b(k,O) + 8ik3 b(k,t') dt

This equation is equivalent to the differential equation

(2.3.3.6), derived by the heuristic argument.

4ik3 - C = 0

t

0

We have thus established:

Theorem 2.3.2. Let the potential u(x,t) satisfy t h e conditions

of theorem 2 . 3 . 1 , and let X = k2 be any point of the continuous

spectrum. The reflection coefficient b(k,t) is given by

8ik3t = b(k,O)e

2.4. SUMMARY AND DISCUSSION OF THE METHOD OF SOLUTION BY THE

INVERSE SCATTERING TRANSFORMATION.

Collecting the main results of the preceding sections (and

omitting for the moment various conditions to which we shall

return shortly) we arrive at the following mathematical struc-

ture :

We consider the problem of determining the solution u(x,t) of

the Korteweg-de Vries equation

(2.4.1) ut - 6uuX + uxXx = o

with given initial data

(2.4.2) U(X,O) = u(x)

Associated to that problem we consider the SchrGdinger equation

2.4.3) v - (u(x,t) - Alv = 0 x E(-m,m) xx

For t = 0 one can compute the spectrum, which consists of a

finite number (possibly zero) of discrete eigenvalues X = -knl

and a continuous part X = k2. One can further compute the

2


normalization coefficient Cn(0) and the reflection coefficient

b(k,O), both defined in section 2.1.

By the Theorem 2.2.1 the spectrum is invariant with time,

while by Theorems 2.3.1 and 2.3.2 the evolution of Cn(t) and

b(k,t) is given as follows:

4kit (2.4.4) Cn(t) = Cn(0)e

8ik3t b(k,t) = b(k,O)e

The potential of the Schrgdinger equation can be recovered

from the scattering data at any t > 0 by solving the inverse scattering problem. For that purpose we introduce the function

-knC m

(2.4.5) B(5;t) = N 2 I: Cn(t)e + J b(k,t)eikcdk -m n=l

and the Gel'fand-Levitan integral equation

(2.4.6) K(x,y;t) + B(x+y;t) + B(z+y;t)K(x,z;t)dz = 0

In this equation x and t are parameters.

m

X

We obtain the solution of the initial value problem for the

K.d.V. equation from the formula

(2.4.7) a ax u(x,~) = -2- K(x,x;~)

We note that the original problem for the non-linear partial

differential equation (2.4.1) is transformed and reduced in

this way to the problem of solving a one dimensional linear

integral equation.

The Gardner-Greene-Kruskal-Miura procedure, summarized above,

and further developments originating from the G.G.K.M. dis-

covery, are usually called the method of inverse scattering

transformation.

We now summarize the conditions which have been introduced at

various steps of the preceding analysis.

For the theory of the scattering problem we have required, by


condition (2.1.2):

For the invariance of the spectrum we have needed the condition

that

for p = 1,2,3 is bounded for 1x1 -+ to.

Finally, for the results concerning the evolution of Cn(t) and

b(k,t) we have required that

u(xrt) ux(xrt)

tend to zero as 1x1 +. m, uniformly with respect to t on any

compact time interval.

The theory is therefore consistent in a time interval t c[O,T]

in which the solution u(x,t) satisfies these requirements.

However, one can assure the consistence a priori by imposing suitable decay conditions on the initial data uo(x). This

follows from Cohen (1979). From that paper (taking into account

corrections introduced in the Addendum to the paper) we quote

the following results

Assume that uo(x) is three times continuously differentiable

on IR, and has a piecewise continuous fourth derivative.

Assume further that, for 1x1 + m

(2.4.8)

with > y, and y a number to the specified shortly.

Let, fo r any positive number y, the symbol [y] denote the largest integer strictly less than y, and [ O ] = 0.

One now has the following estimate for 1x1 * m, and t in any

compact interval on the positive axis:

INVERSE SCATTERING TRANSFORMATION 2 9

The number y can generically be taken equal to 8 . In an ex-

ceptional case one must take y = 10. This exceptional case is

the case in which the Schrbdinger equation with potential

u,(x) has for X = 0 a nontrivial bounded solution.

It should be clear from the estimates given above that taking

a sufficiently large one can assure the consistence of the

theory on arbitrary compact time intervals. There is further

certainly no reason to worry about consistence if uo(x) decays

exponentially for 1x1 -+ a, or is a function with a compact .

support.

2 . 5 . THE PURE N-SOLITON SOLUTION.

Suppose that the function uo(x), that defines the initial

condition for the K.d.V. equation, is such that the reflection

coefficient b(k,O) is zero. Then, by Theorem 2.3..2, the

reflection coefficient b(k,t) is zero for all time and the

Gel’fand-Levitan integral equation becomes an equation with a

degenerate kernel. Before discussing the solution in that case

let us convince ourselves that there exist large families of

reflectionless potentials uo(x).

The easiest example is provided by the solitary wave solution

of the K.d.V. equation. We recall (eg. (1.2.15)), the formula:

(2.5.1)

After a trivial shift along the x-axis we have, as initial

condition for the K.d.V. equation

( 2 . 5 . 2 )

and hence the Schrbdinger equation

2 uo(x) = -#c sech [%Ex1

30 W. ECKHAUS E A. VAN HARTEN

2

dx + {#c sech2(#fix) + A3v = 0 2 (2.5.3)

We get rid of the free parameter c (the speed of the solitary

wave), by the transformation

(2.5.4) ;;=#fix

which produces

(2.5.5) - dLV + 12 sech2x + i}v = 0 dx2

- 4 with A = -A . Equation (2.5.5) can be analysed in terms of hypergeometric

functions (see for example Morse and Feshbach (1953)).

C

There is only one discrete eigenvalue

(2.5.6) x = -1

and the reflection coefficient is zero.

-

Returning to eq. (2.5.3) we see that for c E(0,m) we have a

one parameter family of reflectionless potentials, with

corresponding discrete eigenvalues

C X = A = - - 1 4' (2.5.7)

We now rapidly extend the families of reflectionless potentials

by a result of Deift and Trubowitz (1979). Reformulating some-

what for our purpose, we have:

Lemma 2.5.1. (Adding a d i s c r e t e e i g e n v a l u e l .

L e t t h e f u n c t i o n qN(x) b e s u c h t h a t t h e S c h r a d i n g e r e q u a t i o n

has N d i s c r e t e e i g e n v a l u e s

X = -k$ n = l,...,N

.kn+l > k., > 0 Vn.


L e t B be any number s a t i s f y i n g

For any such number B one can c o n s t r u c t a f u n c t i o n qN+l(x)

s u c h t h a t

has N+l d i s c r e t e e i g e n v a t u e s , g i v e n by

2 X = X = -knl n = l,...,N n

L = -B

Furthermore, if t h e r e f Z e c t i o n c o e f f i c i e n t f o r qN is z e r o , t h e n

t h e r e f Z e c t i o n c o e f f i c i e n t f o r qN+l is a l s o z e r o .

The proof given in Deift and Trubowitz ( 1 9 7 9 ) is constructive.

This leads to :

Corollary. Given any f i n i t e s e t of n e g a t i v e numbers one can

c o n s t r u c t a r e f Z e c t i o n Z e s s p o t e n t i a Z f o r t h e S c h r d d i n g e r

e q u a t i o n , w i t h t h e s e numbers a s d i s c r e t e e i g e n v a t u e s .

We now turn to the Gel’fand-Levistan equation with reflection

coefficient equal zero. From eqs. ( 2 . 4 . 5 1 , ( 2 . 4 . 6 ) , ( 2 . 4 . 7 ) ,

and suppressing for notational simplicity the time dependence,

we have :

( 2 . 5 . 3 ) K(x,y) + 2 cne + J Z Cne K (x,z) dz=O. n= 1 x n=l d dx ( 2 . 5 . 9 )

N -kn(x+y) m N -kn(z+y)

U(X) = - 2 - K(x,x).

Equation ( 2 . 5 . 8 ) can be written as follows

N -kny -knx ( 2 . 5 . 1 0 ) K(x,y) + Z C,e {e + +,(X)l = 0

n= 1

where

-k On(x) = e K(x,z)dz.

X


-kmY Multiplying eq. ( 2 . 5 . 1 0 ) by e and integrating we obtain,

for m = I , . . .,N : - (kn+km)x -k x

{e ++,(XI) = 0. N 2 1 ( 2 . 5 . 1 1 ) Qm(x) + f: Cn k,+k, e n= 1

In ( 2 . 5 . 1 1 ) we have a system of N linear algebraic equations

for the unknown $ l , . . . , ~ N , which can be solved by standard

procedures. Substituting the solution into ( 2 . 5 . 1 0 ) and per-

forming the operation ( 2 . 5 . 9 ) leads to the final result. In

GGKM ( 1 9 7 4 ) one can find the demonstration of an elegant final

formula, which reads as follows:

( 2 . 5 . 1 2 ) u = -22 log{det(I+C))

where I is the identity matrix while C is the matrix given by

a 2 ax

- ( km+kn ) x ( 2 . 5 . 1 3 ) c [c m c n 1 km+kn e 1 .

Of interest is the asymptotic behaviour of the solution for

large time, where we expect the SOlitOnS to emerge. To

perform the analysis one must reintroduce time evolution of

the coefficients Cn, given in ( 2 . 4 . 4 1 , and study the solution

( 2 . 5 . 1 2 ) in moving coordinates systems

- ( 2 . 5 . 1 4 ) x = x - ct, c > 0.

In the next section we shall study, as an exercise, in some

detail the case N = 2 . For the general case we summarize

here the results of a rather elaborate analysis given in

GGKM ( 1 9 7 4 ) . Similar results have also been obtained by

Zakharov ( 1 9 7 1 ) ,Wadati and Toda ( 1 9 7 2 ) , and Tanaka ( 1 9 7 2 ) .

Let us write

( 2 . 5 . 1 5 ) u(xtct,t)= u(x,t)

and consider

2 ( 2 . 5 . 1 7 ) c = 4kn, n = 1 ,..., N. GGKM(1974) show that, for x in any compact [-XIXI one has


2 2 (2.5.18) lim u(x,t) = -2kn sech [kn(x-E, ) ]

t-- P

where E, is a number that can be computed explicitely. P

Furthermore, for

(2.5.19) c f 4ki, n = l,...,N.

(2.5.20) lim C(F,t) = 0.

Thus, from a reflectionless potential, for which the 2 2 SchEdinger equation has N discrete eigenvalues -kl,...,-kN

there emerge, for t -f m, N solitons, with soliton speed

given by

t+m

2 n c = 4 k .

Similarly, considering the behaviour of u(x,t) given by

(2.5.12) for t + -m, GGKM (1974) find for c = 4ki :

2 2 lim i(x,t) = -2kn sech kn(z-f 1 1

P (2.5.21)

t+-m

where 5

For c # 4kn one has

(2.5.22) lim u(Fly) = 0. t + - m

This result is a full confirmation of the observations of

Zabusky and Kruskal (1965) described in section 1.3 :

again are numbers that can be computed explicitely. P

2

The N - s o l i t o n s s t a r t o u t for l a r g e n e g a t i v e t i m e s a s N

s o l i t a r y waves. As t h e t i m e advances i n t h e p o s i t i v e d i r e c t i o n

t h e N - s o l i t o n s undergo i n t e r a c t i o n s , from u h i c h t h e y emerge,

f o r l a r g e p o s i t i v e t i m e , unchanged i n shape. The o n l y e f f e c t

of i n t e r a c t i o n i s a s h i f t i n position.

We conclude this section with an interesting representation

formula, due to GGKM(1974).

Lemma 2.5.2. If u is a r e f l e c t i o n Z e s s p o t e n t i a Z t h e n

u = - 4 Z k Q2, where Q N

n= 1 a r e t h e e i g e n f u n c t i o n s c o r r e s p o n d i n g

n n n


2 to t h e e i g e n v a l u e s X = -kn.

For the proof we return to the basic equations ( 2 . 5 . 9 ) - ( 2 . 5 . 1 1 ) . The equations can be written in different form, making use

of the relation

This relation follows from the theory of chapter 4 . Thus we

obtain

-k x N d m ( 2 . 5 . 2 4 ) u ( x ) = 2 Z C [e q ~ , ( x ) l . m= 1

Furthermore, from ( 2 . 5 . 1 1 1 , for m = l,...,N,

-k x -k x N - knx z - m ( 2 . 5 . 2 5 ) Cme = $,(XI + c,e

We rewrite ( 2 . 5 . 2 4 ) as follows:

u = 2(-A+B) N

m= 1

-k x m JIm

( 2 . 5 . 2 6 ) A = 2 Cmkme

N -k,X Wm - dx B = z Cme

m= 1

Multiplying ( 2 . 5 . 2 5 ) by kmqm and summing we obtain

N N N CnCm - (kn+km) x m= 1 m = l n=l kn+km kme JIn'm. ( 2 . 5 . 2 7 ) A = km$i + -

Similarly, multiplying by dJIm and summing we get

Ckn+km)x dJI, JIn ZE *

( 2 . 5 . 2 8 ) e m= 1

We obtain another interesting relation by differentiating

( 2 . 5 . 2 5 ) , multiplying -Jim and summing. This yields N N CnCm - (kn+km) x

JInqm. ( 2 . 5 . 2 9 ) A = -B + 2 2 Z - k +km km m=l n=l n

We now use ( 2 . 5 . 2 9 ) to eliminate the double sum from ( 2 . 5 . 2 7 ) . The result is


This result proves the lemma.

Remark. One can generalize the representation formula given

in lemma 2.5.2 to a class of potentials with non-zero

reflection coefficients. Assuming that kb(k)E L one has 1

* m 2 N

(2.5.31) u = -4 Z kn$i + I kb (k)$z dk. k= 1 m

where b" (k) = b(-k) . Proof of this result can be found in Deift and Trubowitz (1979).

2.6. THE PURE 2-SOLITON SOLUTION: AN EXERCISE.

We consider, as initial condition for the K.d.V. equation

(2.6.1)

This is a reflectionless potential, with two discrete eigen-

values

2 u,(x) = -6 sech x.

(2.6.2) x1 = -1, x2 = -4.

The solution of the K.d.V. equation, obtained through the

procedure of the inverse scattering transformation (GGKM (1974)) reads :

3+4 cosh(2x-8t) t cosh(4x-64t) 3 cosh(x-28t) + cosh(3x-36t) 1'- (2.6.3) u(,x,t) = -12 {

It seems hardly probable that, without the knowledge of

section 2.5, one could predict at a first glance at the

formula (2.6.3) the emergence of solitons. This is why

we propose an exercise in explicite asymptotic analysis.

We introduce moving coordinates

- 0

(2.6.4) x = x - ct + x

and write


From (2 .6 .3) w e o b t a i n

(2.6.5) A - - u ( x , t ) = - 1 2 - B

w i t h

A=3+4~0sh[ 2 (x-x,) + ( 2 ~ - 8 ) t] +cash[ 4 (x-x,) + ( 4 ~ - 6 4 ) t] ( 2 . 6 . 6 1

2 B = { ~ c o s ~ [ ( X - X , ) + ( C - ~ ~ ) ~ ~ + C O S ~ [ 3 (x -xO)+(3c -36) t] 3 -

W e c o n s i d e r t h e behav iour o f A and B f o r t + + and a l l c E (0 ,m) . Both f u n c t i o n s A and B have t h e s t r u c t u r e

Pi (c) t Caie

W e have drawn i n a diagram t h o s e exponents o c c u r i n g i n A and B t h a t a r e impor t an t f o r t h e a sympto t i c a n a l y s i s . The r e a d e r shou ld v e r i f y t h a t t h e exponents n o t i n c l u d e d i n t h e d iagram are, f o r a l l c , dominated by some exponent i n c l u d e d i n t h e diagram. The diagram is t h e key t o t h e a n a l y s i s .

I t fo l lows t h a t i f c f 4 and c +16, t h e r e i s , f o r a l l c , an e x p o n e n t i a l f u n c t i o n i n B t h a t dominates a l l e x p o n e n t i a l f u n c t i o n s i n A . There fo re

- - ( 2 . 6 . 7 ) l i m u ( x , t ) = 0 f o r c # 4 , c # 1 6

t ++ Take now:

(2.6.8) c = 4 .

S t r a i g h t f o r w a r d a n a l y s i s shows:

- 4 (X-x,) e - -

( 2 . 6 . 9 ) l i m u ( x , t ) = -6 - t + + m ‘(x-x,) - 3 (X-x,) l2

{+ Te +

3 (x-x,) = -6f7e



We recover the symmetric formula of the solitary wave by

chosing:

(2.6.10) x0 - l 1 n 3 - - 2

Similarly:

which becomes the standard expression for a solitary wave when

one takes

1 1 (2.6.12) xo = 7 In 3.

Repeating the analysis for c = 16 one finds:

-2 (2.6.13) lim ;(x,t) = -8Icosh(2;) 3 t+Tm

where one must take

We see from the results that the speed of the emerging

solitary waves is given by the formula

2' (2.6.15) c = -4X.with X = X1 and X = X

2 . 7 . RELATION BETWEEN SOLITON SPEED AND EIGENVALUES.

In sections 2.5 and 2.6, for the case of pure N soliton solu-

tion, a relation between the velocities of the emerging soli-

tary waves and the discrete eigenvalues of the Schrzsainger equation was


e s t a b l i s h e d by a n a l y s i s o f t h e e x p l i c i t formulas f o r t h e s o l u t i o n . I t t u r n s o u t t h a t t h i s r e l a t i o n a l so h o l d s i n a much more g e n e r a l s e t t i n g , i . e . f o r s o l u t i o n s o f t h e K.d;V. e q u a t i o n which a r e n o t r e f l e c t i o n l e s s p o t e n t i a l s fo r t h e SchrSdinger e q u a t i o n . T h i s r e s u l t is r a t h e r fundamental and i s due to Lax(1968) . W e fo rmula t e it as f o l l o w s :

Theorem 2 . 7 . 1 . L e t u ( x , t ) be a s o l u t i o n of t h e K.d.V e q u a t i o n

wh ich i s un i formZy bounded f o r t E [ O , a ) .and s a t i s f i e s t h e

c o n d i t i o n s of t heorem 2 - 2 . 1 . Suppose & h a t t h e r e e x i s t s a

number c > 0 s u c h t h a t

l i m u (X+c t -xo , t ) = u (X,c) t + m

u n i f o r m l y f o r 1x1 Q X , where X i s a n a r b i t r a r y number, and

U ( x , c ) i s a s o Z i t a r y wave s o l u t i o n o f t h e K.d.V. e q u a t i o n

moving w i t h s p e e d c .

Then

c = -4x P

where A i s a d i s c r e t e e i g e n v a l u e of P

vxx - [ u ( x , t ) - A I v = 0 .

Comments on t h e theorem.

L e t us i n t r o d u c e a t r a n s f o r m a t i o n to moving c o o r d i n a t e s

- ( 2 . 7 . 1 ) x = x - c t + xo

and w r i t e f u r the remore

u(X+ct-x t) = U ( X ; c ) + w ( X , t ) . 0 ’ ( 2 . 7 . 2 )

The SchrSdinger e q u a t i o n becomes

Cons ider on t h e o t h e r hand t h e Schrcd inge r e q u a t i o n


We know from section 2.5 that there is one discrete eigenvalue

(2.7.5)

The proof of the theorem would be an elementary exercise in

spectral perturbation analysis, if we were given that w(x,t)

tends to zero as t + a, uniformly on the whole x-axis. This

however is not

t -+ m, is only

the whole ;-ax

other solitary

that makes the

shall proceed

of its own.

the case. The convergence of w(z,t) to zero, as

on compact intervals, and cannot be extended to

s , because at large distances on the ;-axis

waves may be present. It is this circumstance

proof of the theorem a non-trivial exercise. We

n a number of steps, each having some interest

Lemma 2.7.1. C o n s i d e r t h e e i g e n v a l u e p r o b l e m

vxx - (U(x-ct)-X)v = 0

where U(x-ct) i s a s o Z i t a r y wave of t h e K . d . V . e q u a t i o n . Then

1 ac A = -

i s an e i g e n v a l u e and

# IJJ = C(-U)

t h e c o r r e s p o n d i n g e i g e n f u n c t i o n . C i s a n o r m a Z i z a t i o n c o n s t a n t .

Proof of Lemma 2.7.1. The assertion concerning the eigenvalue

was already demonstrated in section 2.5 using the explicit

formula for U and explicit results about the Schrbdinger

equation. One can further prove the lemma without such explicit

results, using instead the relations that define U from section

1.2.

After substitution x = x-ct we compute -I L

U- X

U-- 1 xx (2.7.6) 1

Using (1.2.12) and (1.2.131, with m=n=O, we obtain


1 1 (2.7.7) - JI,, = --[-U2- +I (4)

and finally

(2.7.8) $JEE 1 - [U(Z)-Al$l = (-U) 4 1 ( h + p )

It is clear that $ satisfies the SchrGdinger equation when

A = - ~ c , which proves the lemma. 1

We introduce now, for notational convenience, the operators Lt

and Lm, which have as their domain the dense subset of L (IR)

given by {v E L (IR) Iu E L (IR)), as follows:

2

2 2 2

dx2

(2.7.9) L v = - - dG2

dLv

dx (2.7.10) Lmv = -q - U(G)V

where U(G) is again the solitary wave of the K.d.V. equation.

We shall demonstrate:

Lemma 2.7.2. Let u(x,t) be as specified in Theorem 2.7.1, which

impZies that the function

w(G,t) = u(x+ct-xo,t) - U G )

satisfies

lim w ( G , t ) = o , 1x1 E x t+m

+ for any number X EIR

eigenfunction given in Lemma 2 . 7 . 1 , corresponding t o the

discrete eigenvaZue A o of

. Let further $' be a real normalized

0 (L,+A )v = 0

Then:

II (Lt+A 0 0 )I) II Q 6(t)ll$ 0 II

2 where 1 1 . I( is the norm of L ( IR) and 6(t) is c1 positive con-

tinuous function such that


lim 6(t) = 0 t+m

Proof of Lemma 2.7.2

It should be clear that

(2.7.11)

We write

(2.7.12)

m

~~(LttA0)J10112 = 11wJIol12 = I w 2 [ J I 0 2 - 1 dx. -m

-X - 2 0 2 m 2 0 2 - 2 0 2 -

Iw [ J , 1 dx = I w [ J , I dx t Iw [ J I I dx t IW~[J,~I~&. -m -X X -m

We consider the first integral on the right hand side. By

elementary estimates we have

with

B(x,t) = max w 2 - (x,t). (2.7.14) IxlQ

Now

lim B(X,t) = 0 for each X E IR, . t+m

Using an asymptotic extension theorem (Eckhaus (1979)) it

follows that there exists a positive monotonic function Xo(t)

with

(2.7.15) lirn Xo(t) = m.

Such that

t+m

Thus we can write


From the explicit formula for the eigenfunction JIo which follows from lemma 2.7.1 we have that for lzl sufficiently large there exists a constant A such that

-2JXOX (2.7.18) [$'I2 G A e

Furthermore, w2 is uniformly bounded. These facts lead to :

(2.7.19) ~ ~ [ $ ~ l ~ d G < B(X0(t),t)llJI 11 + Ce 0 2 m

-m

where C is some constant.

Now, given (2.7.15) , (2.7.16) and the trivial observation that

we deduce that

m

(2.7.20) J w2[$'l2dX < 62(t)ll$0112 --a,

where 6(t) is some positive continuous monotonic function,

with

(2.7.21) lim 6(t) = 0. t-tm

This proves the lemma. We now proceed to :

Proof of theorem 2.7.1

We shall need the following basic fact from the spectral

theory (see for example Kato (1966)) :

Let L be a densely defined selfadjoint operator in a Hilbert

space. If h is not in the spectrum of L then there exists a

constant K such that, for all functions v is the domain of definition of L

(2.7.22) II (L+X)vll Kllvll.

Furthermore, the existence of a positive constant K for which

inequality (2.7.22) holds is a necessary and sufficient

condition for X to be outside the spectrum of L.


Now consider an operator Lz that depends on a parameter z , and

suppose that one has an upper bound

(2 .7 .23) K f f ( 2 )

with f ( z ) such that

(2 .2 .24) lim f ( z ) = 0 . Then one must have

Z'Zo

(2 .2 .25) lim dist(X,S(LZ)) = 0

where S ( L z ) is the spectrum of Lz. This follows from the fact

that, given ( 2 . 7 . 2 3 ) , ( 2 . 7 . 2 4 ) , for z = z there does not

exist a positive constant K such that ( 2 . 7 . 2 2 ) would be

satisfied, and X I for z = zo,musttherefore be in the spectrum

Z+Z0

0

of Lz.

With these preparations we turn to the theorem. Suppose that

X o , given in ( 2 . 7 . 5 ) , is not in the spectrum of Lt. Then

there would exist a constant K such that, for all v in the domain of L

(2 .2 .26)

t

0 II(Lt+ h )vll 2 Kllvll

Furthermore by lemma 2.7.2 we have an upper bound

with lim 6 (t) = 0. This implies that t+m

lim dist(XO,S(Lt)) = 0. t--

However, by the theorem 2 . 2 . 1 the spectrum of Lt is invariant

with time, while X o is a constant. We arrive at a contradiction, and must conclude that X o is a point of the spectrum for all time. Since XO = -%c is negative, x is a

discrete eigenvalue of Lt which proves the theorem.

0

From the theorem 2.7.1 one can further deduce :


Corollary. Suppose t h a t t h e S c h r a d i n g e r e q u a t i o n , w i t h p o t e n -

t i a l u'(x,t) s a t i s f y i n g t h e Korteweg-de h i e s e q u a t i o n , h a s N

d i s c r e t e e i g e n v a l u e s . Then t h e r e a r e a t m o s t N numbers c s u c h

t h a t , i n a r b i t r a r y compacts 1x1 d,VX,

lim u(x+ct-xo,t) = u(X;C). t+m

In other words, from arbitrary initial conditions, corre-

sponding to N discrete eigenvalues for the SchrGdinger

equation, at most N solitary waves of the K.d.V. equation can

emerge. However, we cannot yet conclude that, in accordance

with one's expectation, the number of solitary waves

emerging from arbitrary initial conditions will be equal to

the number of discrete eigenvalues, as was the case for the

reflectionless potentials treated in section 2.5.

2.8. THE EMERGENCE OF SOLITONS FROM ARBITRARY INITIAL

CONDITIONS.

Now we turn to the general case, i.e. initial conditions for

the K.d.V. equation which give rise to N discrete eigen-

values of the SchrGdinger equation, with N # 0, and a

reflection coefficient b(k) 9 0. Because the solitary waves

of the K.d.V. equation move to the right, while the dispersive

waves move to the left, and furthermore 'the solitons move

faster when their amplitude is larger, one should expect for

large positives times the emergence of N solitons, each

followed by a decaying tail of dispersive waves, the solitons

being arranged into a parade with the largest one in front.

(Miura (1976) 1 . Several attempts have been made to demonstrate

this conjecture (Segur (19731, Ablowitz and Segur (1977)). F u l l and rigorous demonstration has been given only recently in

Eckhaus and Schuur (1980). The demonstration is achieved by a

rather simple abstract analysis, complemented by a considerable

amount of h.ard explicit computations and estimates. We shall

describe the reasoning in this section and refer frequently for

technical details to the publication mentioned above, which

will be denoted by ES for short.


2.8.1 Formulation of the problem.

We shall use a slightly different form of the Gel'fand-Levitan

equatiom. Introducing in the standard form of the equation,

given in section 2.1, the transformation of variables

(2.8.1.1) y = 2y*+x, z = 2z*+x

One obtains after some trivial manipulations, and dropping the

stars on the variables at the end, the equation

(2.8.1.2) @(y;x,t) + R(x+y;t) + J S?(x+y+z;t)@(z;x,t) dz = 0

with y 0

m

0

8ik't (2.8.1.6) b(k,t) = bo(k)e

(2.8.1.7) C.(t) = C.(O)e . 4k3t

3 7

The unknown @(y;x,t) is a function of the variable y; in

the integral equation (2.8.1.2) x and t are parameters. The

solution of the K.d.V. equation is given by

a + (2.8.1.8) u(x,~) = - a , B ( O ;x,t).

We shall study the solution of (2.8.1.2) in moving coordinates

in the parameter space x,t, defined by

2 - (2.8.1.9) x = ~ - 4 c t, VC E R+.

In particular, we shall study the behaviour for large positive

times, with x confined to arbitrary compacts independent of t. For each c = k we expect to see a

soliton emerging.

< M, and EI

i


Now we give the problem an abstract formulation:

Let V be the Banach space of real continuous functions, bounded

for y E ( O , m ) , and equiped with the supremum norm.

For each g E V we define the mappings

(2.8.1.10)

(2.8.1.11)

m

(Tdg) (y) = I Oa(x+y+z;t)g(z)dz

(Tcg) (y) = I Rc(x+y+z;t)g(z)dz. 0 m

0

T clearly is a mapping of V into V; T will be investigated in

the sext subsection. d C

Our problem is thus to find an element B E V such that

(2.8.1.12) (I+Td)B + TcB = -Q

Q = R + R d c (2.8.1.13)

where I is the identity mapping.

We know the solution 8, of

(2.8.1.14) (I+Td) 8, = -nd

which yields the pure N-solution of the K.d.V. equation. We

intend to study the full problem as a perturbation of the pure

N-soliton case.

2.8.2 Analysis of Rc and Tc.

We consider

m 2 2 (2.8.2.1) Rc(x+4c - 2 t+y;t) = - 1 Jb (k)e 2ik (x+y) ,8itk (c +k ) dk

*-Ca 0

It should be clear that Qc is an oscillatory integral for large

t, 1x1 M, and tends to zero as t -+ m. The precise behaviour

depends on .the behaviour of bo(k), which in turn is determined

by the initial condition for the K.d.V. equation.

Imposing suitable conditions on bo(k) one can derive estimates

of the type


2 (2.8.2.2)

with o(t) + 0 as t -+ m , and F(y) bounded and integrable on the

positive y-axis, under the condition that 1x1 G M.

lQc(x+4c t+y;t) I < F(y)U(t)

For example, suppose that bo(k) is analytic in 0 < Im(k) < E ,

where E is arbitrarily small positive number, and bo (k) = o (2 ) for Ikl + m uniformly in that strip. Then

where y and c1 are positive constants. Demonstration, which is

an exercise in the complex plane, is given in ES.

Similarly, if one assumes bo(k) to be p) 2 times differentiable,

and satisfying together with its derivatives suitable integra-

bility and decay conditions for Ikl + m, then an estimate of the

type (2.8.2.2) follows using essentially integration by parts.

The decay factor a(t) is in this case algebraic.

Finally, very similar estimates can be obtained for the anc

ax derivative y, which will be needed later on in the analysis.

For technical details of .these estimates the reader can consult

ES . With the result (2.8.2.2), examplified by (2.8.2.3), we proceed

to investigate the mapping T . In moving coordinates we have: C

2 m

(2.8.2.4) (Tcg) (y) = I Qc (x+4c t+y+z; t) g (z) dz. 0

Hence m

Thus finally,


where A i s some c o n s t a n t . E x p l i c i t l y , i n t h e a n a l y t i c case g iven i n (2 .8 .2 .3 ) , w e f i n d

Thus w e have e s t a b l i s h e d t h a t T i s a con t inuous mapping o f V i n t o V , and t h e norm o f Tc t e n d s t o ze ro for t + QJ and 1x1 G M .

C

2.8.3 S o l u t i o n of t h e Gel ' fand-Levi tan equation.

W e cons ide r

(2 .8 .3 .1) ( I + T d ) B = - (Q+TcB) . The o p e r a t o r I + T d r e p r e s e n t s an i n t e g r a l e q u a t i o n w i t h a degene ra t e k e r n e l . Therefore , s o l u t i o n s of

can be s t u d i e d e x p l i c i t l y . I n ES, by an e x t e n s i v e e x e r c i s e i n l i n e a r a l g e b r a and a n a l y s i s o f l i m i t s f o r t + m , it is shown t h a t t h e i n v e r s e ( I + T d )

V i n t o V , and fur thermore t h a t t h e i n v e r s e is uni formly bounded f o r t -f m , provided a g a i n t h a t 1x1 M.

-1 indeed e x i s t s as a mapping of

To s i m p l i f y t h e n o t a t i o n w e s h a l l w r i t e

-1 (2 .8 .3 .3 ) ( I + T d ) = S

and w e have

Thus w e can " i n v e r t " (2 .5 .3 .1) and o b t a i n t h e e q u a t i o n

(2 .8 .3 .5) B = -SR - STcB. 5

Now c o n s i d e r t h e mapping T, d e f i n e d by

z

(2 .8 .3 .6) Tg = f - STcg; f , g E v.


By t h e r e s u l t s ( 2 . 8 . 3 . 4 ) and ( 2 . 8 . 2 . 6 ) w e have

(2 .8 .3 .7 )

Hence, f o r s u f f i c i e n t l y l a r g e t , one h a s llST 11 < 1, and T i s a c o n t r a c t i v e mapping i n t h e Banach s p a c e V. I t f o l l o w s t h a t a unique s o l u t i o n g o f

IISTcII < IISlI .IITcII < A a a ( t ) .

- C

(2 .8 .3 .8 ) g = f -STcg, f , g E V

e x i s t s . Fur thermore , one e a s i l y o b t a i n s a n estimate f o r t h e s o l u t i o n as fo l lows :

This y i e l d s

2.8 .4 Decomposition of t h e s o l u t i o n and estimates.

We w r i t e

(2 .8 .4 .1 ) B = Bd + B c

w i th

(2 .8 .4 .2 ) B d = -SRd.

2 E x p l i c i t e a n a l y s i s g iven i s ES shows t h a t Bd(y:z+4c t , t l i s uni formly bounded f o r t E [ O , m ) , 1x1 Q M I y E (0,m). W e recal l t h a t f3 produces t h e pure N-so l i ton s o l u t i o n of t h e K.d.V. e q u a t i o n through t h e formula

d

- + - 2 ( 2 . 8 . 4 . 3 ) u d ( x l t ) = -- a Bd(0 ;x+4c t , t ) . ax I n t r o d u c i n g t h e decomposi t ion ( 2 . 8 . 4 . 1 ) i n t o (2 .8.3.5) w e have t h e e q u a t i o n

(2 .8 .4 .4 ) 8, + STcBc = -SRc - STcBd.

From t h e a n a l y s i s of t h e p r e c e e d i n g s u b s e c t i o n w e know t h a t a unique s o l u t i o n Bc e x i s t .


To estimate t h e s o l u t i o n w e p roceed as fo l lows :

Our f i n a l r e s u l t a t t h i s s t a g e is t h a t i n a l l moving coordinaks x = x - 4c t , vc E IR+ , i n any compact I X ~ G M , f o r large t

(2.8.4.7) IIB,II = O ( a ( t ) ) .

2 -

Furthermore, i n t h e f i r s t approximat ion w e have

(2.8.4.8)

We reca l l t h a t i f t h e r e f l e c t i o n c o e f f i c i e n t b o ( k ) is a n a l y t i c i n a s t r i p 0 QIm(k)Q E, E > 0 then u ( t ) = O (e-"t) , a > 0.

2 = -S(Rc+T B ) + O(0 ( t ) .

BC c d

Unfo r tuna te ly t h e l abour is n o t f i n i s h e d y e t . The s o l u t i o n of t h e Korteweg-de V r i e s e q u a t i o n i s g iven by

- - + - 2 (2 .8 .4 .9 ) u ( x , t ) = Ti,(x,t) - 1 a B C ( O ;x+4c t , t ) . ax

Thus w e need estimates of t h e d e r i v a t i v e o f B, w i t h r e s p e c t t o x . To o b t a i n t h e s e estimates w e r e t u r n to t h e e q u a t i o n (2.8.4.4) and d i f f e r e n t i a t e bo th s i d e s w i t h r e s p e c t t o x. Denoting t h e d e r i v a t i v e s by pr imes we f i n d :

-

Using s u b s e c t i o n 2 .8 .3 w e conclude aga in t h a t a unique s o l u t i o n 6; e x i s t s , and proceed t o estimate t h e s o l u t i o n . AS o u t l i n e d i n s u b s e c t i o n 2.8.2, it is n o t d i f f i c u l t t o estimate QA, which immediately l e a d s t o estimates o f Th. W e a l r e a d y have t h e e s t i m a t e s f o r f3, and B d , w h i l e e x p l i c i t e a n a l y s i s shows t h a t 6; i s uni formly bounded f o r t , y E [ 0,-) I 1x1 Q M .

However , t h e estimate of S ' r e q u i r e s a n o t h e r e x t e n s i v e e x e r c i s e i n l i n e a r a l g e b r a and t h e a n a l y s i s o f l i m i t

52 W . ECKHAUS & A. VAN HARTEN

behav iour , t h e r e s u l t o f which i s t h a t S' can b e shown t o t h e uni formly bounded f o r t -+ m, 1x1 < M.

Thus w e a r r i v e a t t h e f i n a l r e s u l t , which can b e summarized as fo l lows :

S o l u t i o n s u ( x , t ) o f t h e Kortweg-de V r i e s e q u a t i o n t h a t evo lve from a r b i t r a r y i n i t i a l data u,(x) r a p i d l y f o r 1x1 * f o r t h e whole o f t h e t h e o r y to h o l d ) , when viewed i n moving c o o r d i n a t e s x = x + 4c t , f o r any c > 0 ,

and i n any compact 1x1 < M , f o r t -+ m are g i v e n by:

(which decay s u f f i c i e n t l y

2

2 (2 .8 .4 .11) u (x+4c t , t ) = ud(x , t ) + O ( a ( t ) )

where u d ( X I t ) is t h e pu re N-so l i ton s o l u t i o n , and N is t h e number o f d i s c r e t e e i g e n v a l u e s c o r r e s p o n d i n g t o t h e p o t e n t i a l u,(x) . The f u n c t i o n a ( t ) t e n d s t o z e r o as t + m, t h e e x a c t behaviour of a ( t ) depends on p r o p e r t i e s o f t h e r e f l e c t i o n c o e f f i c i e n t b o ( k ) . I f b ( k ) i s a n a l y t i c i n a s t r i p 0

0 t h a t s t r i p , t h e n u ( t ) t e n d s t o z e r o e x p o n e n t i a l l y .

2 Im(k) Q E, E > 0 , and b o ( k ) = o ( k ) f o r k -t m un i fo rmly i n

CHAPTER 3

ISOSPECTRAL POTENTIALS

THE LAX APPROACH

Let L be a family of closed operators on some Banach space of

functions V, and let L have the structure

o u L = L + M

where Lo is some fixed operator, and MU is the multiplication

by a family of functions u(x,t), with t a parameter.

For any fixed t, the spectrum of L is the collection of all

values of X for which the operator

L + A

does not have a bounded continuous inverse on all of V.

For the sake of a convenient terminology we shall call u(x,t)

isospectral potentials if the spectrum corresponding to u(x,t)

is invariant with t. One of the main discoveries in the GGKM-

analysis was the fact that solutions of the Korteweg-de Vries

equation with suitable decay properties for 1x1 + m are iso-

spectral potentials for the Schrbdinger equation. This dis-

covery naturally leads to (at least) three basic questions:

I.

I1 a

Are there other equations, then the K.d.V. equation, of

which the solutions are isospectral potentials €or the

Schrbdinger equation?

Are there other eigenvalue problems, then the Schrbdinger

equation, for which one can find isospectral potentials as

solutions of some interesting evolution equation?

111. Given an evolution equation for functions u(x,t) can one

find an eigenvalue problem for which u(x,t) are iso-

53


spectral potentials?

P. Lax (1968) answered (affirmatively) question I, and

developed a formalism which pointed the way to answer question

11. Question I11 is at the present date essentially not

answered in any systematical way.

The Lax approach starts with the observation that the pheno-

menon of two operators having the same spectrum is a wellknown

one in the theory of selfadjoint operators in a Hilbert space,

and is connected with the concept of unitary equivalent

operators. In this setting Lax developed a formalism which

permits a characterization of isospectral potentials. We shall

follow the reasoning of Lax (19681, filling in some mathema-

tical details, in section 3.2. We commence however in a much

simpler setting in section 3.1, and restricting ourselves to

the discrete eigenvalues we derive the main result by an

elementary analysis. In sections 3.3 and 3.4 we follow again

Lax (1968). Finally, in the last section, we take a different

point of view, drop the restriction to selfadjoint operators,

and generalize in various directions the results of 3.1 and

3.2.

Let us begin with some technical remarks. In the spectral

theory one is often led to consider operators L which are not

defined for all elements of a Hilbert space V, but only for a

dense subset Vo C V. For example, in the case of the

Schrgdinger equation, it is natural to consider the spectral

problem in the Hilbert space of squared integrable functions

L2( IR) , while Lv only makes sense for elements of L ( IR) that

have their first and second derivative in L (IR).

2

2

Operators of which the domain is a dense subset of a Hilbert

space V are called densely defined in V. In what follows we

shall always assume that all operators occurring in our

analysis are defined on some common dense subset Vo of V. This

statement will not be repeated at all occasions.

We further note that there may exist values of X in the

THE LAX APPROACH 55

spectrum such that the equation

(L+X)v = 0

has nontrivial solutions which are not elements of the Hilbert

space V. Such values usually constitute the continuous part of

the spectrum. In the case of the Schrgdinger equation the

corresponding nontrivial solutions are oscillatory functions

which are not squared integrable. These generalized eigen-

functions lie in a larger space v‘ 3 V.

As a final preliminary we introduce the notion of the derivat-

ive of a family of operators, parametrized by t, with respect

to t, in a way parallel to classical definition of derivatives

of functions

Let F(t) have a domain Vo c V and range in a Banachspace

W . F(t) is differentiable at t = to if for all v E Vo

F (tO+A) -F (to) v exists in W. A 1 im

A+ 0

The derivative of F(t) at t = t is an operator 0

(y ) t=tO

such that, for all v E Vo

F ( to+A ) -F (to )

A V. A+O

For example’, in the case of the Schrgdinger operator L, onehas

aL - = MU at t 3u(x,t) , where M is the multiplication by the function at

Ut

We consider now two families F1 (t) , F2(t) and the product operator Fl(t)F2(t). Differentiation of the product is not a

trivial matter, as one discovers when attempting to prove the

product rule. Furthermore, it is not difficult to give examples

in which Fl(t) and F2(t) are differentiable, but Fl(t)F2(t) is

not.


However, we can establish the following result:

Lemma 3 .1 . . Let: F2 (t) have a domain W6W,F1 (t) have a domain V 6 V and

range F2 (t)CVO, where V is a Hizbertspace. Let F1 (t) ,F2 (t) and F1 (t) IF2 (t) be

d i f ferent iable

mZe hoZds, i . e .

aF2

aF1 (t) aF2 (t)

and moreover range K(t) C VO. Then the usual product

F2(t) + Fl(t) at. - F (t)F2(t) = at a at 1

Proof. For any v E Vo we consider the identity

F1 (to+A)F2(t0+A)-F1 (t0)F2,(t0) =

F1 (tO+A) -F1 (to)

A

- - A F2(t0) v +

F2(tO+A)-F2 (to) V + F1 (tO+A) A

On the left hand side the limit for A + 0 exists and equals

The limit of the first term on the right hand side also exists

and equals

aF (t) [71+t=to F2 (tO)V.

Therefore the limit of the second term on the right hand side

also exists. However, we cannot assert yet that the limit of

that product operator equals the product of the limits of the

two operators.

To circumvent the difficulty we proceed as follows:

Let denote the inner product on our Hilbert spaceV. We

consider, for v E v ,W E Vt the expression 0

F2 (tO+A) -F2 (to) A V,W) = J(A) = ( F1 (tO+A)

w i t h F; the adjoint of F1, which has a damin C V (dense).

THE LAX APPROACH 57

We may now pass to the limit and have

This shows that, in weak sense

However, we already know that the limit above also exists in

strong sense, while strong and weak limits, if they both exist, are equal. This proves the lemma.

Comments. In applications the conditions of the lemma can be

verified using some further information about the problem

under consideration. For example consider a one parameter

family of operators L and let c(t) be a. family of eigenvalues,

with $(*,t) the corresponding family of eigenfunctions, i.e.

L$ + <$ = 0

Suppose that one can show (as in the case of the SchrBdinger

equation, in Chapter 4 ) , that 5 (t) and $ ( *,t) are continuously

differentiable with respect to t. Then LJI also is continuous-

ly differentiable. If (by inspection) L is continuously diffe-

rentiable, then, using the lemma, one has indeed

a. L$ = 9 $ + L 2. at at

Another result in the spirit of Lemma 3.1, which will also be

useful to us in this chapter, is given in:

Lemma 3 . 2 . Let Fl(t) be continuous in t and bounded and F2(t)

differentiable. If F1 (t) is differentiable then F1 (t)F2 (t) is

differentiable and the usual product rule holds. Conversely,

if one is given that Fl(t)F2(t) is differentiable, then Fl(t)

is differentiable, if range F2(t) is equal to Vo.

The proof is left as an exercise to the reader.


3.1. THE INVARIANCE OF DISCRETE EIGENVALUES BY AN ELEMENTARY APPROACH

In t h i s section we derive the following results:

Theorem 3.1.1. Let L be a one parameter f a m i l y (parame t r i zed b y

t) of s e l f a d j o i n t opera tors dense ly d e f i n e d on a H i l b e r t space

V and cont inuouszy d i f f e r e n t i a b l e w i t h r e s p e c t t o t. Suppose

t h a t t h e d i s c r e t e e igenva lues of L are c o n t i n u o u s l y dz ' f f e ren -

t i a b l e w i t h r e s p e c t t o t, and t h a t t h e same ho lds f o r t h e corresponding e i g e n f u n c t i o n s $, w i t h $, at a$ E Vo. Suppose

f u r t h e r t h a t t h e r e e x i s t s a one parameter f a m i l y o f opera tors

B such t h a t

aL a t - = BL - LB

aL w i t h B, BL, LB, L, at dense ly d e f i n e d on a common subse t o f

vo c v.

Then t h e d i s c r e t e e i g e n v a l u e e o f L a r e i n v a r i a n t w i t h t.

Furthermore, i f an e igenvazue is s imp le , t h e n t h e corresponding

e i g e n f u n c t i o n Jl s a t i s f i e s t h e e v o l u t i o n equa t ion

.?!k = (B+C)$ a t where c is an a r b i t r a r y cont inuous f u n c t i o n o f t. I f B + C is an t i symmet r i c t h e n YJll i s independent o f t.

Proof. Let ~ ( t ) be a family of eigenvalues and Jl(.,t) the

corresponding family of eigenfunctions. From

(3.1.1) LJl + SJl = 0

differentiating with respect to t, we obtain

We introduce

(3.1.3) a L a t - = BL - LB

This yields

(3.1.4) [L+Clg + BLJl - LBJl + JI = 0

THE LAX APPROACH 59

From (3.1.1) we also have

(3.1.5) BLJI = -5BJI

So that (3.1.4) finally becomes :

Let ( * I - ) denote the inner product in V. We take the inner

product of the function on the left hand side of (3.1.6) with

the eigenfunction J I . This leads to:

(3.1.7) a l l $ I 1 2 at = -( JI,[L+c] (g - BJI)) We now use the fact that L is selfadjoint, and obtain

(3.1.8) g!l$Il2 = -( (L+~)IJJ~ $ - BJI) The right hand side is zero by virtue of (3.1.1). Hence

(3.1.9) 2 L O at

This proves that if 5 is an eigenvalue for some t = to then 5

is an eigenvalue for all t. Hence the eigenvalues are invariant

with t.

We now return to equation (3.1.6). Using (3.1.9l1 we have:

This equatiowis satisfied if

(3.1.11) g - BJI = CJI

where C is an arbitrary function of t. From that equation it

also follows that

Thus, if B+C is antisymmetric, then

a (3.1.13) - l I J 1 1 1 2 = 0

This concludes the proof of the theorem.

at


Remarks.

It should be clear that the evolution equation for eiqenfunc-

tions is determined in the last stage by the normalization of

eigenfunctions. Thus if one has chosen I I I J J I I = 1, and B is anti-

symmetric, then C = 0. However, it may be convenient to define

eigenfunctions in a different way, as will be the case in

chapter 4 . Then the norm of IJJ depends on t and the evolution equation takes a different form.

This remark is of particular importance if one considers the

case of an eiqenvalue with multiplicity > 1. Solving (3.1.6) one obtains for each of the eigenfunctions $iI i = l , . . . . , u r

the equation

a ~ . u ‘ 1

(3.1.14) - = BQi + C cijIJJj at j=l

A careful “normalization” of the functions $ I ~ , for example by

prescribing their behaviour at infinity, is needed in order

to determine the functions cij.

3.2. THE INVARIANCE OF THE SPECTRUM.

We shall now, in a more abstract setting, derive the following

result:

Theorem 3.2.1. Let L be a one parameter famiZy of selfadjoint

operators (parametrized by t) denseZy defined on a HiZbert

space V, and continuously differentiable with respect to t.

The spectrum of L is invariant with t if there exists a one

parameter family of antisymmetric operators B such that:

i. - = aL BL - LB at

ii. The operator equation

with (U)t=o = 1 (the identity operator) hasas solution a

one parameter family of operators on v for all t > 0. iii. LU is differentiable with respect to the parameter t.

THE LAX APPROACH 61

We shall establish the result in a number of steps. We first

look for pairs of operators L and L which have the same spec-

trum. This search can be facilitated by the use of the so-

called unitary operators. We recall the definition (Yoshida

(1974) ) :

-

A bounded linear operator U on V is called unitary if

Range (U) = V and U is isometric, i.e.

(3.2.1) (uv,uw) = (v,w) vv,w E v

where ( .,.) is the inner product on V. We next define unitary

equivalence of operators, as follows:

Two selfadjoint operators L and L on V are unitary equivalent

if there exists a unitary operator U such that

5

5

(3.2.2) U-lLU = L

We now have:

- Lemma 3.2.1. If two seZfadjoint operators L and L are unitary

equivaZent then they have the same spectrum.

We sketch the proof:

Consider a value of X that is not in the spectrum of z. The < + A has a continuous inverse, and the solution of

(Z+X)v = f

v = (Z+X)-lf

is given by

Consider for the same value of X the problem of solving

(L+X)w = f

With the aid of U-lL = ZU” we transform the problem into

(Z+X)U-lw = U-lf

and obtain the solution

w = u(E+x)-lu-lf


From this one can conclude that X is not in the spectrum of L.

Repeating the reasoning for a value of X that is not in the

spectrum of L leads to the conclusion that the resolvent sets

of L and L are the same. Furthermore, a direct and elementary

demonstration shows that the discrete eigenvalues of L and

are the same.

-

We observe now that starting with an arbitrary unitary operator

on V (for example the identity operator) and an antisymmetric

operator B (which may depend on a paramater t) one can con-

struct a one parameter family of unitary operators, under the

solvability assumption ii. stated in the theorem. This is

formalized in:

Lemma 3.2.2. L e t u be a one p a r a m e t e r farniZy of o p e r a t o r s ,

p a r a m e t r i z e d by t, which s a t i s f i e s

- = BU at

where B is a n a n t i s y m m e t r i c o p e r a t o r f i . e . t he adjointB*=-B). U is

u n i t a r y f o r a 2 2 t if (U)t,O is u n i t a r y .

Proof. We consider any pair of functions v1,v2 E V and the

corresponding one parameter families of functions

(3.2.3) w1 = UVl ; w2 = uv2

Restricting v1,v2 to a dense subset

in the domain of B, we have

of V such that w1,w2 are

aw2 BW2 Bw2 ; at = awl

at - = (3.2.4)

We now compute

(3.2.5) aw2 ) = ( w ,Bw2) = ( B * w1,w2) = 4 BW rw2) = ( W1 'at 1 1

= -(-,w2) 3% at

Thus

a -(w ,w ) = 0 at 1 2

THE LAX APPROACH

It follows that

63

(3.2.6) (w1,w2) = (UVl‘UV2) = (Uvl,Uv2)t=0 = (Vl,V2).

This proves that U is isometric. From the result (3.2.6) we

further deduce that (U*Uvl,v2) is independent of t for all

v1,v2 in a dense subset of V, which in turn implies that U*U

is independent of t. Thus:

(3.2.7) u*u = (U*U)t,O = 1

which proves that U is unitary (Yosida (1974)).

We now turn to the proof of the theorem. From the relation

(3.2.8) u-lu = I

when U is differentiable, one can establish, using Lemma 3.2,

that U - l is differentiable. Therefore, any operator L, unitary

equivalent to L and given through

(3.2.9) L = u LU also is differentiable (Lemma 3.2, using boundness of U-l).

We consider

-

-1 -

a - a LU = - UL. at at (3.2.10) -

L is selfadjoint, which permits to use Lemma 3.1. We thus have - au au aL a L u + L - = - L + u - at at at at (3.2.11) -

we are given that U evolves according to

(3.2.12) - - - BU at

Introducing this into (3.2.11), eliminating by (3.2.9), and

rearranging we obtain

(3.2.13)

5

aL aL at at (- + LB - BL)U = U -

the left hand side is zero by condition i. of the theorem,

theref ore


- (3.2.14) - aL = 0 at - Thus L is independent of t and consequently the

- spectrum of L

also is independent of t. By lemma 3.2.1 the unitary equivalent

operators L and L have the same spectrum. This concludes the

proof of the theorem.

*

3 . 3 . ISOSPECTRAL POTENTIALS FOR THE SCHRODINGER EQUATION

In order to test the usefulness of the results of the preced-

ing sections we consider the SchrGdinger equation. In that case

we have

(3.3.1) u (x, t) L = - - a2

ax 2

2 on the usual Hilbert space L (IR) with the inner product

(3.3.2) (w,v) = .f G d x W

-W

a The operator at L, in theorems 3.1.1 and 3.2.1 becomes the

multiplication operator -ut, i.e. Vv E V,

- Lv = -u v. at t a

The procedure of application of Theorems 3.1.1, 3.2.1 consists

essentially of two steps:

I.. Find an (antisymmetric) operator B such that

( 3 . 3 . 3 ) BL - LB = \

where Pb is the operator of multiplication by w, with

W = K(u) i.e. Vv E Vo, BLv - LBv = K(u) v.

11. For each such operator B, families of isospectral potent-

ials u(x,t) are defined as solutions of the equation

If one uses theorem 3.2.1 then in a final stage the solvabili-

ty condition ii. of that theorem must be verified. However,

THE LAX APPROACH 6 5

in many practical applications it may be sufficient to use

theorem 3.1.1 which asserts the invariance of the discrete

eigenvalues, and establish the invariance of the continuous

part of the spectrum by other means. Such is the case for the

Schrgdinger equation, where the invariance of the continuous

spectrum follows from the prescribed behaviour of the poten-

tials at infinity (see chapter 4 ) .

Following Lax (1968) one can search for the operators B system-

atically by investigating families of real linear differential

operators. If B is required to be antisymmetric, then the

differential operators must be of uneven order, and must

furthermore be of the structure

where b are at this stage unknown, while q is an arbitrary

integer, or zero. j

We start with q = 0, i.e.

- a Bo - ;i-;; (3.3.6)

straightforward computation yields

Thus BOL - LBO in this case indeed is a multiplication operator, The isospectral potentials u(x,t) satisfy

au au at ax - = - (3.3.8)

However, the result is trivial: solutions of ( 3 . 3 . 8 ) are

functions

(3.3.9) u(x,t) = U(x+t)

hence, a transformation of variables

(3.3.10)

produces a time-independent SchrGdinger equation.

E = x + t


Hoping for a less trivial result we consider next

(3.3.11) - a3 a a + b l a x + a x b l ax

B1 - - 3

We have to compute now

a3 a a a2 ax ax

(3.3.12) BIL - LB1 = (7 + bl ax + bl) (7 - U) -

Performing the exercise one obtains, after some labour,

au abi a2 ax ax ax (3.3.13) BIL - LB1 = - ( 3 - + 4 -) - - 2

a’bl au - (7 a u + 2b2 ax + - 3 ) ax ax

3

We require that BIL - LB1 be a multiplication operator. This means that the differentiations must disappear. We take

3 bl = -p

and obtain

Isospectral potentials u(x,t) thus satisfy

- 6uuX) 1 (3.3.15) Ut = T(uxxx

This equation (modulo some trivial transformations) is the

Korteweg de Vries equation!

The invariance of the spectrum of the Schrgdinger equation for

potentials satisfying the K.d.V. equation (Theorem 2.2.1) is

thus demonstrated by the Lax procedure, in a way essentially

different from the G.G.K.M.-analysis.

One can furthermore procede to study the differential operators

B for q > 1. The requirement that B L - LB be a multipli- q q 9

THE LAX APPROACH 67

cation operator determines the unknown coefficients of the

operator B As a result one obtains isospectral potentials for

the SchrGdinger equation as solutions of "higher order K.d.V."

equations. The computational labour naturally increases with q.

The reader may find it interesting to study in this way for

example the case q = 2 .

q'

We shall finally investigate the consequences of the evolution

equation for eigenfunctions derived in section 3.1. i.e.

(3.3.16) JIt = BJI

We develop this result further for the Korteweg-de Vries

equation. First we modify somewhat the operator B1 to arrive at

the standard form of the K.d.V. equation, and take

This produces the evolution equation for eigenfunctions

2 n Let now X = -k be a discrete eigenvalue and J, the correspon-

ding normalized eigenfunction. Then, from the SchrGdinger

equation, we further get

- (3.3.19) JIxxx -

Combining (3.3.18)

(3.3.20) JIt = 2

(u-h)JIx + uxJ,

and (3.3.19) leads to

which is in complete agreement with the results of section 2.3.

3.4. ISOSPECTRAL POTENTIALS FOR MORE GENERAL SELFADJOINT

OPERATORS

Summarizing the procedure described in section 3.3 and genera-

lizing at the same time the class of operators L to which it

is applied, leads to the following wellknown result of Lax

(1968) :


Theorem 3.4.1. L e t L be a s e l f a d j o i n t o p e r a t o r d e n s e l y d e f i n e d

on a H i l b e r t space V , and l e t L have t h e s t r u c t u r e

L = Lo + MU where Lo i s i n d e p e n d e n t of u and Mu i s m u l t i p l i c a t i o n by u.

Suppose t h a t t h e r e e x i s t s an a n t i s y m m e t r i c o p e r a t o r B, s u c h

t h a t

K (u) BL - LB = M

w i t h B, BL, LB d e n s e t y d e f i n e d on V . Then t h e e i g e n v a t u e s o f L

a r e i n v a r i a n t w i t h t i m e f o r a l l f u n c t i o n u(x,t) s a t i s f y i n g

ut = K(u)

Comments. We can prove the theorem either in the setting of

section 3.1, or in that of section 3.2. Thus if we assume that

the eigenvalues and eigenfunctions of L are continuously

differentiable with respect to time, then the theorem is a

direct consequence of theorem 3.1.1. On the other hand if we

assume that the operator equation

with (U)t,o = I has as solution an operator U in V for all

t > 0, and LU is differentiable with respect to t, then the Theorem 3.4.1 immediately follows from Theorem 3.2.1. Further-

more, in that case we have the invariance of the whole

spectrum.

As an example of application Lax (1968) takes for u a symmetric

p x p matrix and for L the matrix operator

then find a third order matrix operator B satisfying the

conditions of the theorem. This yields isospectral potentials

as solution of the matrix X.d.V. equation

+ tu. One can ax

1 u + +ux + u u) + uxxx = 0 t X (3.4.1)

Lax (1968) further states, and we quote:

rrOther c h o i c e s of t h e o p e r a t o r L s h o u t d l e a d t o o t h e r c l a s s e s

of e qua t i o n s

THE LAX APPROACH 69

This remark led the way in an important breakthrough in the

development of the method of inverse scattering transformation,

given in Zakharov and Shabat (1972),(1973). These authors have

shown that one can find a pair of operators L and B satisfying

the conditions of theorem 3.4.1 such that the isospectral

potentials u(x,t) are solutions of

u = i(uxx + 2u2E) t (3.4.2)

where ii denotes the complex conjugate of u. Equation (3.4.2) is called the nonlinear SchrBdinger equation and plays an

important role in various wave propagation phenomena (see for

example Whitham (1974)). Solution of the nonlinear SchrGdinger

equation by the method of inverse scattering transformation

was the first demonstration that the method is not essentially

limited to equations of the Korteweg-de Vries family. Further-

more, the scattering problem introduced by Zakharov and Shabat,

and generalizations of that problem, have led to important

further developments. We state here the generalized Zakharov-

Shabat problem in a nonselfadjoint form, which is best suited

for the purpose of the inverse scattering theory. One then has,

for the pair of functions v1 (x) , v2 (x) , x E ( - m , m ) the eigen-

value problem

avl

av2 ax

- - a x ~2 = -Xvl

+ rvl = -Xv2 - - -

where q and r are potentials. The choice 1: = -q , q = u links equation (3.4.3) with the non-linear SchrGdinger equation. The

theory of the Zakharov-Shabat scattering problem will be given

in Chapter 5, while further developments based on that problem

will be described in Chapter 6.

3.5. AN ALTERNATIVE APPROACH

In this section we develop a line of reasoning different from

Sections 3.1 and 3.2, which will lead to generalizations of the

the results.


We shall deal with operators satisfying the following:

Conditions on L .

i . L i s a one parameter f a m i l y of l i n e a r o p e r a t o r s d e n s e l y

d e f i n e d o n a H i Z b e r t space V and c o n t i n u o u s l y d i f f e r e n -

t i a b l e w i t h r e s p e c t t o t h e parameter t.

and moreover:

ii . For any number X t h a t beZongs t o t h e s p e c t r u m of L t h e r e

e x i s t s a ( g e n e r a l i z e d ) e i g e n f u n c t i o n v, eZement of a l a r g e r space V' 3 V, which s a t i s f i e s

LV + xv = 0

iii. ConverseZy , i f for some number X one can f i n d an e l e m e n t

v E V' 3 V such t h a t

LV + Xv = 0

t h e n 1 i s i n t h e s p e c t r u m of L.

At t = 0 we single out some arbitrary but fixed number < = X

that belongs to the spectrum of L and denote by +(xlO) a

corresponding eigenfunction, solution of

(3.5.1) LIJJ + A$ = 0

Now we let the function v(x,t) evolve according to

( 3 . 5 . 2 ) - BV ; v(x,O) = IJJ(x,O) av at - -

where B is a linear densely defined operator, which is such

that a unique solution of (3.5.2) exists and is in V' for a l l t > 0.

Next we define a function f (x,t) I by

( 3 . 5 . 3 ) f(x,t) = LV(X,t) + Xv(x,t)

and differentiating we obtain

af = aL v + L - av + h- av at at at at (3.5.4)

In doing so we implicitly assume that v is in the domain of

THE LAX APPROACH 7 1

av at at and - in the domain of L.

We further have the initial condition

(3 .5 .5 ) f(x,O) = 0

Using the evolution equation ( 3 . 5 . 2 ) we find

af aL at at - = -V + LBV + ABV

Furthermore, from (3.5.3)

( 3 . 5 . 6 ) BLV + ABV = Bf

Hence finally

aL at at

f(x,O) = 0

af - Bf = (- + LB - BL)V ( 3 . 5 . 7 )

Again we naturally assume that all operators occurring in

( 3 . 5 . 7 ) are defined on a common dense subset.

Suppose now that B is such that

( 3 . 5 . 8 )

From the unique solvability of ( 3 . 5 . 2 ) it follows that the

prob 1 em

- aL + LB - BL = 0 at

( 3 . 5 . 9 ) _ - af Bf = 0 at

has only the trivial solution for t E ( O , m ) . Hence f 0 for

all t 2 0. Thus A belongs to the spectrum of L for all t 2 0,

and v(x,t) defined by ( 3 . 5 . 2 ) is a corresponding eigenfunction.

We have thus established:

Theorem 3.5 .1 . L e t t h e o p e r a t o r L s a t i s f y t h e c o n d i t i o n s i . t o i v . L e t A be any p o i n t t h a t b e l o n g s t o t h e s p e c t r u m o f L a t

t = 0 , and $(x,O) a c o r r e s p o n d i n g ( g e n e r a l i z e d ) e i g e n f u n c t i o n ,

e l e m e n t 07 V' 3 v. Suppose t h a t t h e r e e x i s t s , f o r e a c h X a l i n e a r o p e r a t o r B s u c h

t h a t


i . v = Bv , V(X,O) = VJ(Xl0) t

has a u n i q u e s o l u t i o n v(x,t) i n V' f o r t E ( 0 , m ) .

i i . 2 = BL - LB

w i t h L, B, BL, LB and at d e f i n e d on a common d e n s e s u b s e t .

Then any p o i n t X t h a t b e l o n g s t o t h e s p e c t r u m of L a t t = 0 ,

b e l o n g s t o t h a t s p e c t r u m f o r a l l t > 0.

'dL

Comments. In theorem 3 . 5 . 1 L is not restricted to selfadjoint

operators and B not restricted to antlsymmetric operators.

Although one may search (as in sections 3 . 1 , 3 . 2 , 3 . 3 ) for

operators B that satisfy Condition li and are Independent of X I it may also be advantageousto work with X-dependent operators

B. We shall use this procedure in Chapter 6 when considering

a first order differential operator L acting on vector valued

functions. B can then simple be chosen as a matrix.

We note that theorem 3 . 5 . 1 does not yet assure Invariance of

the spectrum of L with t, because at some t > 0 new eigenvalues

can arise. To exclude this possibility heavier requirements on

the operator B must be imposed.

Theorem 3.5.2. L e t t h e c o n d i t i o n s of Theorem 3 . 5 . 7 h o l d , how-

e v e r , l e t B be d e f i n e d i n d e p e n d e n t of X and s a t i s f y f u r t h e r t h e

f o l l o w i n g a d d i t i o n a l c o n d i t i o n :

iii. For e a c h to E ( 0 , m ) and each Vo E V' t h e r e e x i s t s a

u n i q u e "backward" s o l u t i o n of

Vt = Bv I (V)t'tO = v 0

i n V' f o r t E ( - 6 + t O r t 0 ) , w h e r e 6 i s a n a r b i t r a r y s m a l l

p o s i t i v e n u m b e r w h f c h can be c h o s e n i n d e p e n d e n t o f t o .

Then t h e s p e c t r u m o f L i s i n v a r i a n t w i t h t.

Proof. Suppose that a number A * is in the spectrum of L for

t = to > 0, but is not in the spectrum for $r< to. We now solve

( 3 . 5 . 1 0 ) Vt = Bv I (V)t=tO = $*

THE LAX APPROACH 73

where $* is an eigenfunction corresponding to X*. By the con-

dition iii. the solution v(x,t) exists for t E(-G+tO,tO).

Because v(x,t) is continuous with respect to t (and even

differentiable), and v(x,to) = $* (x) is not identically zero,

v(x,t) is not identically zero for all t E(-G+tO,tO). Repeating

the reasoning that gave proof of Theorem 3.5.1 backward in t

one arrives at the conclusion that A * is in the spectrum for t E (-6+t0 , to) , which contradicts the initial assumption, since ItA-tol can be chosen arbitrarily small.

Concluding remarks. 1 ) The extension of the Lax formalism to

not selfadjoint operators L and not necessarily antisymmetric

operators B which may depend on A , described in this section,

has been used in the literature in a formal way, which consists

of the statement that invariance of the spectrum of L and the

evolution equation for the eigenfunctions (3.5.2) are compa-

tible under the condition ii.

2) If the solvability conditions for B imposed in Theorems

3.5.1 and 3.5.2 are satisfied on a finite interval of t only,

then the conclusion of the theorems holds on such interval.

3) The invariance of the spectrum of L can in application be

demonstrated on the basis of Theorem 3.5.1 (and thus without

the heavier conditions of Theorem 3.5.2) if one has certain

additional informations about the problem. Such is the case

if one can demonstrate by other means (explicit analysis, as

for the Schrbdinger equation) that the continuous part of the

spectrum is invariant with t, while the discrete eigenvalues

are continuous functions of t for t E ( o , ~ ) .


CHAPTER 4

DIRECT AND INVERSE SCATTERING

FOR THE SCHRbDINGER EQUATION

The one dimensional Schradinger equation arose in physics with

the birth of quantum-mechanics, in the beginning of this

century (see Schrgdinger, 1926).

It asks for wave functions I$ as non-trivial solutions of

I$” + (A-u)$ = 0, ’ = x E R . ax (4.1)

In this equation the real function u is called the potential

and X is a spectral parameter, which is interpreted as the

energy of the state $ .

However, not all values of the spectral parameter X are physically interesting. If u(x) -)r 0 for 1x1 -+ m sufficiently

rapidly,then only those spectral values X are physically interesting for which a non-trivial solution of 4.1 exists

which behaves as follows :

(i)

(ii) a scattered wave, i.e. $(XI is asymptotically periodic

a bound-state, i.e. I$ E LZ(R )

for 1x1 -+ m ,

Shortly after the introduction of the Schrgdinger equation

mathematicians realized that one of their tasks should be to

develope a spectral theory covering problems such as 4.1. The

development of this mathematical theory did not take long :

already in the fundamental paper of von Neumann, 1929 the

abstract spectral decomposition theorem for self-adjoint

unbounded operators on Hilbert spaces was given, which can

be applied to 4.1:

Around the middle of this century the mathematics for the

75


Schrodinger equation and more generally for self-adjoint

ordinary differential operators, was well-understood, see

Kodaira, 1950, Coddington, Levinson, 1955. A lot of information

especially on the spectral theory of Schrodinger's equation

can also be found in the book of Glazman, 1963

At the same time mathematicians also posed the inverse problem

for the Schrodinger equation : is it possible to reconstruct

the potential from the spectral data and if so what information

about the spectrum does one need and how can the construction

be done? This inverse problem was solved by Gelfand, Levitan

in 1951 and later in a more manageable form my rlarchenko, 1955,

Faddeev, 1959.

In recent days the interest for SchrBdinger's equation and the

theory of inverse scattering became very vivid again, because

of its unexpected relation with the Korteweg-de Vries equation,

see section 2.2.

This new interest has lead to still new discoveries and better

understanding of the theory, see Deift, Trubowitz, 1979.

In this chapter we shall follow an approach to direct and

inverse scattering and spectral theory for SchrGdinger's

equation which is essentially elementary.

We start in section 4.1 by giving definitions and introducing

notation for the solutions of Schrodinger's equation and for

the scattering data. Some elementary properties of these

solutions and scattering data are also presented there.

Next in section 4.2 a number of results will be given

concerning the regularity, dependence on the data and

asymptotic behaviour of the solutions of Schrodinger's

equation.

In section 4.3 we shall consider the relation of the scattering

data to the structure of the spectrum of the Schrodinger operator on L2(R 1 .

We continue in section 4.4 with the derivation of Fourier

integral expressions for the solutions of the SchrBdinger

SCHRbDINGER EQUATION 77

equation. This is a necessary preparation for section 4.5.

Then in section 4.5 we derive the Gelfand-Levitan-Marchenko

integral equation and discuss its unique solvability.

This integral equation is the heart of the inverse scattering

theory. For its derivation we shall proceed in great lines as

in Ablowitz, 1978. The method given there is rather trans-

parant and has the advantage that it can be generalized at

once to the Zakharov-Shabat system of equations (see chapter 5).

Finally we conclude this chapter with the discussion of some

generalizations.

Let us indicate before we go on with the analysis what type of

conditions the potential u will have to satisfy.

We assume that the potential u is a real function which (i) is

sufficiently regular and (ii) satisfies a growth condition

for 1x1 + m. As for the regularity of u we suppose :

The growth of u for 1x1 + m will be restricted as follows

and moreover :

-m

We shall call (4.2,3,4) a growth condition of order m. In all

what follows the value of m will be at most 2 . However, for

quite a number of results it is sufficient if(4.4) is

satisfied with m = 0 or m = 1.

In fact there is quite a discussion in the literature con-

cerning the question what are the weakest conditions on the

growth of the potential u necessary to derive certain (inverse)

scattering results. For example in Faddeev, 1959 only a growth

condition o'f order 1 is supposed. However, as demonstrated in

Deift, Trubowitz, 1979 this requirement is not sufficient for

all of the results given there. The latter authors work with a

growth condition of order 2.


One of the most interesting questions is, where a growth

condition of order 2 is really necessary. We use a growth

condition of order 2 only to show that the transmission coef-

ficient is continuous at k = 0, where X = k2. This fact plays

an important role in the derivation of the Gelfand-Levitan-

Marchenko integral equation.

4.1.SOLUTIONS AND SCATTERING DATA OF SCHRbDINGER'S EQUATION.

Let us start with the introduction of certain families of

solutions of the Schradinger equation. These families will be

parametrized by k, where k represents a square root of the

spectral parameter X , i.e. a number E @ such that

(4.1.1) A = k 2

M For k E $+ (live. 1x11 k the following way :

0) we define a solution $r of (4.1) in

In order to satisfy (4.l)we must then have :

(4.1.3) R" - 2ikR' = uR.

Furthermore we require :

(4.1.4) lim R(x,k) = 1, lirn R'(x,k) = 0.

In section 4.2 it will be shown that the problem(4.1.3-4)for

R has a unique classical solution,if the potential u satisfies

certain growth condition. Hence Qr is well-defined in this way.

X+-m x+-m

Using the uniqueness property of the solution of (4.1.3-4)it is

easy to verify the relations :

(4.1.5)


These relations imply that for k on the imaginary axis and

Im k 2 0 R and JIr are real.

It is also important to notice that for real k, k Z 0 the

functions JIr and 5, are two linearly independent solutions of the Schradinger equation.

(b)

Since the real axis is unbounded to both sides it is logical

to introduce also a solution JI, of (4.1) with prescribed behaviour

€or x -+ + m .

For k E v+ (1.e. exactly as in a!), we define :

where L satisfies

(4.1.7) L" + 2ikL' = UL

(4.1.8) lim L(x,k) = 1, lim L'(x,k) = 0.

The unique solvability of the problem(4.1.7-8)will be demon-

strated in section 4.2. Moreover we can prove relations

analogous to (4.1.5)

X-+W X+m

Consequently L and J IL are real for k on the imaginary axis and Im k 2 0.

For k real, k f 0 we thus find another pair of linearly

independent solutions of the Schr6dinger equation : JI, and $,. -

Note that at this stage we have introduced for k real and # 0

four solutions of equation (4.1) : JIrr JIrr JIk and vt . Let us now exploit the fact that the 1-dimensional Schradinger

equation is a 2nd order ODE : given 2 linearly independent

solutions each other solution can be expressed as a linear

combination of them.

-

80 W . ECKHAUS & A . VAN HARTEN

( 4 . 1 . 1 1 )

where fi-,R+rr+rr- a r e f u n c t i o n s o f k E R \ ( O )

$r = r+qR + r-vR

W e are now a b l e t o d e s c r i b e t h e a sympto t i c behav iour f o r 1x1 + m of Jlll I$lI qr and $; w i t h k E R \ ( O ) f i x e d .

% i k x ( 4 . 1 . 1 2 ) $& ( x r k ) e f o r x -+ +m

a+(k )e ikx+Q_(k)e -ikx fo r x -+ -m

f o r x + +m i k x ( ik) - '$ ( x , k ) % e

-ikx fo r x -+ -a = I l+ (k )e I l - (k )e

$, ( x r k ) = e f o r x + -m

f o r x + +m

( ik) - '$ ; (x ,k) -e f o r x -+ -m

fo r x .+ +m

ikx-

- ikx - ikx = r+ ( k ) eikx+r- ( k ) e

- ikx - ikx

% r+ ( k ) eikx-r- ( k ) e

The c o e f f i c i e n t s R+,!L-lr+,r- s a t i s f y a number of r e l a t i o n s , g iven below :

Lemma 4 . 1 . 1 . L e t k be r e a l and # 0 . Then :

( 4 . 1 . 1 4 ) R+(k) = r - ( k ) , R-(k) = - r + ( k )

(4 .1 .15)

Proof o f lemma 4 . 1 . 1 . The proof o f ( 4 . 1 . 1 3 ) i s an a lmos t t r i v i a l Combination of (4.1.5-9) and (4.1.10-11).

I k+(k) l 2 = I RJk) I 2 + 1, I r - ( k ) l 2 = I r + ( k ) l 2 + 1

The r e l a t i o n s g iven i n 4.1.15 f o l l o w from t h e f a c t t h a t e a c h p a i r o f s o l u t i o n s $ ,$, of SchrBd inge r ' s e q u a t i o n s a t i s f i e s { I J J ~ $ ~ - $ ~ $ ~ ) ' = 0 , i . e . t h e Wronskian ql$; - $;$, i s c o n s t a n t f o r x E R . Hence f o r any such p a i r $1,@2 it h o l d s t h a t :

1

SCHRbDINGER EQUATION a1

When we apply this argumentation to the pairs $. 5. and 2' a

for k E R \ t o ) and we use the asymptotics given in (4.1.12) we find the contents of (4.1.15). U s i n g we obtain :

(4.1.16) &IRL'-LR'+ZikRLI = r-(k) = !L+(k).

Herewith the first relation given in (4.1.14)has been proven.

The second relation of (4.1.14) is found as follows. When we

introduce the asymptotics of $ for x -t m given in

4.1.12)into (4.1.10)we obtain the identity 0 = i-r- + R+:+ for

k E R \ { O ] . Since r- = L+ and 1r-l = l k + l 2 1 division by r-

k r r

in this identity gives the desired result.

A few remarks should be made here. In the first place one

could think that taking x + -m in (4.1.1l.l using (4.1.12) would

lead to even more relations between a+,L-,r+,r, for k E R \ I O ) .

This appears not to be the case. Secondly? we notice that the

expression (2ikI-l {RL'-LR'+2ikRL} I which appears in (4.1.161,

is perfectly defined and independent of x E R for all

k E ?+\ I01 . This opens the way to extend the domain on which r-,1+ are defined from R \ ( O } to v+\{O).

Definition. For k E z+\{Oj we d e f i n e r- and R.+ by

r-(k) = R.+(k) = {RL ' -LR' +ZikRL). def

On R \ { O ] t h i s a g r e e s w i t h what we d e f i n e d b e f o r e .

The fact that r- and f+ have z + \ { O } as their natural domain

will play a very important part further on.

Let us now discuss hriefly the physical interpretation of the solutions $, and qr of schraaingczc's equation. The timedependent quantuin mechanics leads us to consider the functions e

For real k and k > 0 we can interprete these timedependent functions

nicely in physical terms. using (4.1.12) for -+it is clear that e

represents a wave caning f m the left of which an Wlitude fraction

scattered back. ~nalogously e-iXt$r represents a wave

of which an amplitude fraction I l/r-(k) I travels taklards - and an amplitude fraction lr+(k)/r-(k) I is scattered back.

-iXt -iXt JI, and e JIr.

'iXt$

l/k+(k) travels towards +a3 and an amplitude fraction la-(k)/a+(k) I is fm the right


Hence f o r r e a l 1, X > 0 w e have found s c a t t e r e d waves a s s o l u t i o n s o f (4.1)!

I t i s now l o g i c a l t o i n t r o d u c e t h e f o l l o w i n g q u a n t i t i e s f o r k E I R \ { O } :

a = ,yl : t h e l e f t t r a n s m i s s i o n c o e f f i c i e n t

a = r-

bn. - R-8, br - r+r-

R

r

( 4 . 1 . 1 7 )

: t h e r i g h t t r a n s m i s s i o n c o e f f i c i e n t

: t h e l e f t r e f l e c t i o n c o e f f i c i e n t

: t h e r i g h t r e f l e c t i o n c o e f f i c i e n t

-1

- -1

- -1

I n terms o f t h e s e t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s w e can r e f o r m u l a t e ( 4 . 1 . 1 0 - 1 1) as

( 4 . 1 . 1 8 ) an.$, = bRJlr + Tr, k E I R \ { O }

( 4 - 1 . 1 9 ) arJlr = brJln. + vn., k E I R \ { O } .

The r e a d e r w i 1 . l a g r e e t h a t (4 .l. 19) is n o t r e a l l y a d e e p r e s u l t . Y e t t h i s s imple i d e n t i t y w i l l be t h e s t a r t i n g p o i n t o f t h e d e r i v a t i o n o f t h e Gel'fand-Levitan-Marchenko i n t e g r a l e q u a t i o n i n s e c t i o n 4 . 6 !

The t r a n s c r i p t i o n o f lemma 4 . 1 . 1 i n terms o f r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s y i e l d s :

Lemma 4 . 1 . 2 . L e t k b e r e a l and f 0 . Then :

( 4 . 1 . 2 0 ) % ( k ) = % ( - k ) , a r ( k ) = a,(-k)

E r ( k ) = br ( -k ) ar (k) an. (k )

- -

En. (k ) = ba (-k) ,

an . (k ) = a r ( k ) , b n . ( k ) = -- gr (k ) ( 4 . 1 . 2 1 )

v i t h ( a n . ( k ) I > 0

Of c o u r s e one can i n t e r p r e t e t h e r e l a t i o n s i n ( 4 . 1 . 2 2 ) i n p h y s i c a l terms as c o n s e r v a t i o n of ene rgy f o r k E IR+.

SCHRbDINGER EQUATION 8 3

We conclude by remarking that we can extend the domain of

definition of at and a

k's in p+\IO3. quite naturally to certain values of r

Definition. I f k E ?+\{Oj and r-(k) = t+(k) Z 0 at(k: = ar(k) - - r-(k)-' = !L+(k)-'.

t h e n we d e f i n e

de f

4.2. PROPERTIES OF SOLUTIONS.

The reader will recall that we introduced two families of

solutions of the SchrGdinger equation :

with k E E+.

In this section we shall show, that under suitable growth

conditions on the potential u the problems(4.1.3-4)and(4.1.7-8)

for R and L are uniquely solvable. Since the properties of

these solutions are basic for the remainder of the chapter, we

shall study them in detail.

It is clear that the problems for R and L are very similar. In

fact the mathematics for the problem for L is completely

analogous to that of the problem for R. We shall therefore

deal in our proofs with the problem for R only. The proofs in

the case of L are left to the reader as excercises.

The organization of this section is as follows :

First we shall reformulate the problems for R and L as

integral equations in section 4.2.1. In the following section

we shall consider the questions of existence and uniqueness of

a solution of these equations. After that some results on the

regularity of the solutions and their asymptotic behaviour

will be derived in the sections 4.2.3 and 4 . 2 . 4 .

It will appear that in the analysis it makes a great difference

whether we allow k to be equal to zero or not. In the sections

4.2.2, 4.2.3 and 4.2.4 we suppose that k E g+, k Z 0. The subject of section 4.2.5 is the behaviour of R and L near k=O.

a4 W. ECKHAUS & A. VAN HARTEN

Finally, in section 4.2.6, we consider the properties of R and L in the case of a potential u(x,t) depending on a parameter t.

This is of course important for applications to the Korteweg-

de Vries equation.

4.2.1 Reformulation as integral equations.

We have shown that R and L have to satisfy :

(4.2.1.1) (i) R" = 2ikR' + uR (ii)

lim R(x,k) = 1

lim R' (x,k) = 0

x+- m

x+- m

with k E $+.

L" = -2ikL' + UL lim L(x,k) = 1

lim L'(x,k) = 0

X'W

x*-

Consider now a classical solution R of (i). In the following

way we can derive an integral equation for R.

Let xo be a point E IR. Treating uR as an "inhomogeneous" term

we find by an elementary computation :

2ik (x-x~) R(x,k) = R(x0,k) + R' (xoIk). Ee -1}/2ik +

X X + I Eu(y) I e 2ik(z-y)dz}R(y,k)dy, k # 0.

xO Y X

R(x,O) = R(x0,O) + R' (xoIo) (x-x,) + I

If k E v+\ (0 1 and I I u (y) I dy < m then we find by taking the limit xo + -- : (4.2.1.2) R(x,k) = 1 + I G(x,y,k) R(y,k)dy.

The kernel G is given by :

u(y) (x-y) R(y,O)dy. m xO

-m

X

-a0

However, for k = 0 there are complications with this limit

procedure. This shows already that k = 0 is an exceptional

value. It appears to be necessary to put a stronger condition

on the potential u, if k = 0. Let us suppose :

SCHRODINGER EQUATION 85

k) dy

u (y)

X

= u(y) R(y,k)dy, XO+-m -m

lyldy for x Q 0,

, i.e. lim xoR' (xo,k) = 0. This implies that under a

growth condition of order 1 on u we have an integral equation

of the form given in (4.2.1.2)with kernel given by :

x0+-m

which is indeed the limit of G(x,y,k) as given in (4.2.1.3)as

k + 0.

Analogously we find that if k E u+\ {O} and I or if k = 0 and I I u (y) I (1+ I y I )dy < m then amclassical solution

L of (ii) has to satisfy :

(4.2.1.5) L(x,k) = 1 t I H(x,y,k) L(y,k)dy

with

m

lu (y) Idy < m m

-m

m

X

It is even true that the integral equations (4.2.1.2,5) are

equivalent to the original problems for R and L in the

following sense :

Lemma 4.2.1.1 If k E e+\{O} and u s a t i s f i e s t h e g r o w t h

c o n d i t i o n of (4.4)with m = 0 o r if k = 0 and u s a t i s f i e s t h e

growth c o n d i t i o n of ( 4 . 4 ) w i t h m = 1 t h e n we have :

f i ) R is a c l a s s i c a l s o l u t i o n of (4.2.1.1)-(i) * R is c o n t i n u o u s i n x, bounded f o r x + -m and

R s a t i s f i e s (4.2.1.2).


(iil L i s a c l a s s i c a l s o l u t i o n of (4.2.1.1)-(ii) * L i s c o n t i n u o u s in x, bounded f o r x + m and L s a t i s f i e s ( 4 .2 .1.. 5 )

The "+-ll- part of this lemma is left to the reader as an

excercise in which he can test his abilities in showing

contipity and differentiability in x of integrals of the

form S f (x,y)dy. -m

Under the conditions of lemma 4.2.1.1 the excercise leads to

the following useful expressions for R' and L' :

(4.2.1.7) R' (x,k) = S G' (x,y,k) R(y,k)dy

(4.2.1.8) L' (x,k) = I H' (x,y,k) L(y,k)dy

X

-m

m

X

with G',H' the derivatives of G,H with respect to x, i.e.

(4.2.1.9) G' (x,y,k) = u(y)e 2ik (x-y) , H' (x,y,k) = -u(y)e 2ik (yx)

4.2.2. Existence and uniqueness for Im k 2 0, k # 0.

As usual in the theory of differential equations the

reformulation of the problems for R and L as equivalent

integral equations has advantages for the questions of

existence and uniqueness of a solution.

Before demonstrating the main result (theorem 4.2.1) we

introduce some notations.

+ We define function-spaces W d is the space of all functions w(x,k) on IR x E+\{Ol which for each k E @+\I01 are continuous in x on IR and bounded

for x + 2 m. The spaces W- are endowed with the obvious concept

of convergence :

and W- as follows :

+

Vk E C + \ { O l Va E IR lim [sup Iwn-wI (x,k)l = 0 n- x h

<


BY s+ and S- we denote the following classes of kernels : S- is the space of all functions s(x,y,k) on IR x I€? x g+\{O) which are everywhere continuous in (x,y,k) and satisfy an

estimate X + II Is(x,y,k) Idyl G s(x,k) with s E W. +m

Note that G,G' E S- and H,H' E S+!

-

It may seem somewhat surprising, but we shall not need a

concept of convergence in S-.

We further introduce operations * and *,which vaguely resemble convolutions :

(4.2.2.2) ( s * w) (x,k) = I s(x,y,k)w(y,k)dy, s E S-, w E W-.

(S ** W) (x,k) = I s(x,y,k)w(y,k)dy, s E S , w E W .

+

\I

A X

-m m + + X

.. J Note that in this operation k acts as a parameter.

It is not difficult to verify that the operations * and * have the following important properties :

(4.2.2.3) s E S- and w E W' =$ s * w E W- - Is

ls

: w --* w- is continuous. + s E S+ and w E Wt * s * w E W

: w+ --* w+ is continuous.

These notations enable us to write the integra

R and L in short and elegant way :

equations for

(4.2.2.4) (i) R = 1 + G R (ii) L = 1 + H L

It follows from lemma 4.2.1.1 that looking for classical

solutions of mthe problems 4.2.1.1 (i) and (ii) for R and L with k E E+ and .f ~u(x) Idx<mis equivalent to solving (4.2.2.4) (i) and (ii)

in W- and W+!-

We now formulate the main result of this section.


Theorem 4.2 .2 . Suppose t h a t the p o t e n t i a l u s a t i s f i e s t he growth cond i t ion of ( 4 . 4 ) w i t h m = 0.

( a 1 The problem for R has a unique s o l u t i o n i n W-. T h i s s o l u t i o n s a t i s f i e s (4.2.1.1) l i ) i n c l a s s i c a l sense and can be given as a

Neuman S e r i e s

(4.2.2.5) R = Z Gn.

The Grits (4.2.2.6)

-

m

n= 0 a r e determinded i t e r a t i v e l y by :

Go - - 1, Gn+l = G * Gn , n > 0

and s a t i s f y the e s t ima te :

( 4 . 2 . 2 . 7 ) (Gn(x ,k ) I G (n ! ) - l {Uo(x ) / lk l ln

w i t h Uo (XI = I l u ( y ) Idy.

(b ) The problem f o r L has a unique s o l u t i o n i n W . T h i s soZution s a t i s f i e s ( 4 . 2 . l . l ) ( i i ) i n cZassicaZ sense and is given by

X

-m

- +

m

(4.2.2.8) L = z H n n=O V - w i t h Ho - 1, Hn+l = H * Hn, n 2

IHn(x,k)I (n!)-' {Vo(x)/

Vo(x) = I l u ( y ) Idy . m

X

0.

kiln

The proof of t h i s theorem fol lows t h e l i n e o f t h e classical proof of t h e e x i s t e n c e and uniqueness of a s o l u t i o n of Volterra i n t e g r a l equat ion. The on ly complicat ion is t h e unboundedness of t h e i n t e g r a t i o n i n t e r v a l .

W e s h a l l g i v e t h e proof of p a r t 2 of t h e theorem. However, w e f i r s t s ta te t h e fol lowing r e s u l t of which t h e proof is elementary :


Lemma 4.2.2.1. L e t {gn,n 2 0) b e a s e q u e n c e in W-, s u c h t h a t :

m

Then C gn d e f i n e s an eZement of W-

and G * X gn = X G * gnr m n=O

- m

n= 0 n=O m m

rl

GI * gn = GI * 9,. n=O n=O

The proof of this lemma is very easy and we leave it to the

reader.

Proof of theorem 4.2.1. We proof theestimate given in(4.2.2.7)

by induction with respect to n. Of course(4.2.2.7)is satisfied

for n = 0. For n >

Next using (4.2.2.3)

for all n > 0, R =

m

G * R = X G n=O

Hence R defined as

(4.2.2.4) (i) . Suppose now that % -

0 we find :

X

and lemma 4.2.2.1 it is clear that Gn E W- c n= 0

m

Gn E W- and that :

m

in (4.2.2.5)is indeed a solution in W- of

is another solution of (4.2.2.4") in w-. Then v = R - R satisfies :

Define : M(xo,k) = sup (v(x,k) I . xeO


Of course we have M(xo,k) < m,since v E W-. With induction with respect to n one can easily show that for

x xo

Iv(x,k) I QM(xO,k) {Uo(x)/lklln/(n!).

This implies v(x,k) = 0 on (-mlxo]. But x

Hence we conclude that v 0 on R . This proves the uniqueness

E R is arbitrary. 0

in W- of a solution of (4.2.2.4) (i). n

The following results for the derivatives R' and L' are also

interesting. We can rephrase (4.2.1.7-8) as

(4.2.2.9) R' = G' * R, L' = H' * L. V

Hence these derivatives can be given as series

= z H' * Hn. (4.2.2.10) R' = z G' * Gn, L'

These representations for k E z+\{O} are useful furth(

where we consider the asymptotics of these functions.

easy exercise to show, that

m - v m

n= 0 n=O

r on

It is an

x) P + l

The reader will have noticed that the estimates(4.2.2.7-8)

contain singularities of a bad kind at k = 0. However,we shall

see in section 4.5 that the behaviour of R and L at k = 0 is

usually much better than suggested by these estimates.

4.2.3. Regularity for Im k > 0, k # 0.

We have shown so far that the problems for R and L are

uniquely solvable. It is a tradition in the mathematical

theory of differential and integral equations that one next

poses the question how regular these solutions are. Here

especially the regularity with respect to k is interesting.

We shall prove the following results :


Theorem 4.2.3. L e t t h e p o t e n t i a l u s a t i s f y t h e growth c o n d i t i o n

of o r d e r 0 . Then t h e f u n c t i o n s R,R',R", L,L' and L" a r e

(i) c o n t i n u o u s i n x and k on R x (z+\{O))

(ii) a n a l y t i c i n k on fi!+ f o r each x E R . Let us first try to convince the reader of the importance of

this theorem:

The analyticity of R,R',L,L' in k on @+ for each x E R

an indispensable part in our derivation of the Gel'fand-Levitan-

Marchenko integral equation, since we apply complex integration

and Cauchy's residue calculus to formulas containing these

functions (see section 4.5).

An immediate consequence of theorem 4.2.3. (ii) and (4.1.15) i s

the following result.

plays

Corollary to theorem 4.2.3.

R;'.

f u n c t i o n o f k on Q!+ and a, a meromorphic f u n c t i o n of k on $?!+ w i t h

p o l e s i n t h e z e r o ' s of r-. Moreover, r- i s c o n t i n u o u s on

C+\{OI and a, i s c o n t i n u o u s on E+\{O, z e r o ' s o f r-I.

1 - r-l = C o n s i d e r r-(k) = R+(k) ={m{RL' -LR'+2ikRL) and aR = a, - - Under t h e c o n d i t i o n s o f t h e o r e m 4.2.3 r- i s an a n a l y t i c

-

Before we prove theorem 4.2.3 we shall first derive a useful

lemma.

Let us define :

- Wan = {w E C(R x @+) I w is analytic in k on @+ for

each x E W and satisfies :

(4.2.3.1) = sup SUP Iw(x,k)l def kEK x 6

for all compact subsets K C @+ and all a E R 1 .

Of course we endow W-

by the system of seminorms I given in (4.2.3.1).

with the concept of convergence induced an la,K; a E R , K C @+, K compact

Lemma 4.2.2.

of semi-norms g i v e n i n (4.2.3.1) , i n t h e s e n s e t h a t e a c h

(a) Wan i s c o m p l e t e w i t h r e s p e c t t o t h e s y s t e m


, ~ a u h h y _ s e q u s a c e . c o n v e r g e s ) .

f u n c t i o n s a r e a l s o i n W- :

(4.2.3.2) m 2 o

(b) If h E Wan, t h e n t h e f o l l o w i n g

an

- (4.2.3.3) G * h, GI* h, Gk * h, G i * h.

We have f u r t h e r t h e f o l l o w i n g d i f f e r e n t i a t i o n r u l e s :

- ah (4.2.3.4) $(G h) = Gk h + G * - ak

a - ah - ( G I * h) = Gi * h + G' * ak

A

Proof of lemma 4.2.3.1. We shall heavily use the following

well-known facts :

(i) Let (gn;n E N)

functions on a metric space V, which has the Cauchy

property with respect to the supremum norm on V. Then

there exists a unique bounded, continuous function g on

V such that lim gn = g in the sup-norm on V (see Rudin,

1964, theorem 7.12, page 136).

(ii) Let {gn;n E N 1 be a sequence of analytic functions on C+ to a function g. Then g is analytic on $+ and more-

over differentiation and the limit process are inter-

changeable. (see Conway, 1973,Chapter VII , pg. 147). Of course part (a) of the lemma is a direct consequence of

(i) and (ii).

ad 4.2.3.2.

It is clear that h is for each x E R analytic in k

on @+. The other requirements follow easily from Cauchy's

formula :

be a sequence of bounded, continuous

n+m

a m

where y(kO,E) is the contour (zl Iz-k I = € 1 with 0 < E < Im ko.

0


ad 4.2.3.3-4.

We note that G,G',Gk,GL have the following properties : they

are for y Q x and k E p), continuous in x,y,k and for fixed

x,y analytic in k. Further, their absolute values in x,y,k

with y G x and k E $?!+ can be estimated by C(k).(u(y)l

C(k) continuous and positive on @+.

It is easy to verify that one can take for C(k) in the

respective cases : Ikl-',l, #(Im k)-2, (e Im k1-l.

with

It is now clear that for each of the functions mentioned in

(4.2.3.3) the estimates of (4.2.3.1) hold.

- Consider the case G * h. We have :

X

I G(x,y,k) h(y,k)dy = lim -m A+ -m

uniformly in x,k on compacta C

Furthermore :

X

S G(x,y,k) h(y,k)dy = lim A N-+m

where the Riemann-sum converges

compacta c R x p)+.

uniformly in (x,k) on

- Repeated application of (i) , (ii) shows that G * h is in Wan. We also find by interchanging differentiation and limits :

a X X & I G(x,y,k)h(y,k)dy = I ,,{G(x,y,k)h(y,k)}dy -m -m

i .e the first relation of (4.2.3.4) holds.

The other cases can be dealt with in an analogous way. n

With this preparation the proof of theorem 4.2.3 is now not

difficult.

Proof of theorem 4.2.3. ad ii. We use the series represent-

ation for R given in(4.2.2.5): R = Gn, where the G ' s are

defined iteratively as in(4.2.2.6) and satisfy estimates as

in (4.2.2.7). Applying (4.2.3.3) iteratively we find Gn E Wan.

Using the estimates of(4.2.2.7) in combination with part ( a )

of lemma 4.2.3.1 it is clear that R E Wan. Since R' = G' * R

m

n=O

-


because o f (4.2.2.10), (4.2.3.3) i m p l i e s t h a t R ' E W a n .

The d i f f e r e n t i a l e q u a t i o n for R (4.2.1.1. ( i ) ) y i e l d s t h a t

ad i. The proof of ( i) is c o n s i d e r a b l y e a s i e r t h e n t h e proof o f ( i i) , though i t f o l l o w s ana logous l i n e s . It i s l e f t as a n e x c e r c i s e t o t h e r e a d e r . 0

The r e a d e r w i l l have n o t i c e d t h a t w e d i d n o t u s e t h e f u l l c o n t e n t s o f lemma 4.2.3.1 i n t h e proof o f theorem 4.2.3 hereabove. From (4.2.3.2) it f o l l o w s t h a t a l l d e r i v a t i v e s w i t h r e s p e c t t o k o f R , R ' and R " are e l emen t s of W a n .

Analogous supplements o f theorem 4.2.3 can be g i v e n f o r L .

4.2.4. Asymptot ic behaviour .

U n t i l now w e have p a i d much a t t e n t i o n t o r a t h e r a b s t r a c t p r o p e r t i e s o f R and L and t h e e x p l i c i t behaviour o f R and L

was no t s t u d i e d i n d e t a i l . I n t h e fo l lowing theorem t h i s s h o r t - coming i s mended.

Theorem 4.2.4. Suppose t h a t u s a t i s f i e s a 0 t h - o r d e r g r o w t h

c o n d i t i o n . One t h e n h a s :

- ( a ) a s y m p t o t i c s f o r Ikl + m.

The s e r i e s R = z G n r L = X - . H n r R' = z G ' * Gn and w m m

n=O n=O n= 0 m

L' = z H ' * Hn (4.2.2.5-8-10) r e p r e s e n t c o n v e r g e n t a s y m p t o t i c n= 0

e x p a n s i o n s f o r Ikl -+ m, where t h e n - t h t e r m h a s o r d e r Ikl-" and t h e t e r m s u p t o N a p p r o x i m a t e w i t h o r d e r Ikl , u n i f o r m l y

i n x E R .

-N- 1

( b ) a s y m p t o t i c s f o r 1x1 + m

The Z i m i t s p r e s c r i b e d in(4.2.1.1)for R , R ' , L and L' a r e u n i f o r m

i n k on compacta C

As f o r t h e l i m i t s in t h e n o n - p r e s c r i b e d d i r e c t i o n s we f i n d i f

k Eg,:

@+\{O}.


T h e s e l i m i t s i n ( 4 . 2 . 4 . 1 ) a r e u n i f o r m i n k on compacta c @+. Proof o f theorem 4 . 2 . 4 . The c o n t e n t s o f ( a ) are a d i r e c t consequence of t h e e s t i m a t e s g i v e n in(4.2.2.7-8-13. T h i s i s a l s o t r u e f o r t h e f i r s t p a r t o f ( b ) .

I n o r d e r t o prove t h e second par t o f ( b ) w e f i r z t n o t e t h a t I R ( x , k ) I

T h i s i m p l i e s f o r I m k > 0 and x > 0 :

exp(A/I k l ) on IR x '@+\ (01) w i t h A = I l u ( y ) Idy. -m

I t i s now clear t h a t l i r n R ' ( x , k ) = 0 un i fo rmly on compacta c @+ X+m

W e a l s o have : R L ' , LR ' and R

un i formly i n k on compacta C

AS a consequence :

r - ( k ) = l i m [ ZIT;{RL'-LR'}(X 1

X+m

L-1) t end t o 0 f o r x + m

@ + *

k ) + R{L- l I (x ,k ) + R ( x , k ) ]

= l i r n R ( x , k ) X'm

un i fo rmly i n k on compacta C @+. 0

L e t u s c o n t i n u e w i t h a n impor t an t remark : t h e formula r - ( k ) = l i m R ( x , k ) , IQ k > 0 gives us anotkr veryusefu l

c h a r a c t e r i z a t i o n of r - ( k ) f o r I m k > 0 . T h i s c h a r a c t e r i z a t i o n can be used t o deduce t h e f o l l o w i n g r e s u l t .

X+m

Lemma 4 . 2 . 4 . 1 . Let; u s a t i s f y a 0 - t h o r d e r g r o w t h c o n d i t i o n .

For k E FA{O}

( 4 . 2 . 4 . 2 ) r - ( k ) = 1 - - I u ( y ) R(y ,k )dy .

we have t h e r e l a t i o n : m

1 21k -m


Proof of lemma 4 . 2 . 4 . 1 . For k E (2, w e f i n d , u s i n g t h e i n t e g r a l equa t ion f o r R :

r - ( k ) = l i m R(x ,k) = X+m

2 ik (x-y) -1 ) R (y ,k 1 dy 1 l X = l i m t l + -I u ( y ) (e

X+m 21k -m

dY - 2 1 m k (x-y) X X

S ince \ I u ( y ) e 2ik(x-y)dyl I lu(y)le

we o b t a i n l i m IS u ( y ) e 2ik(x-y)dyl = 0 (compare t h e e s t i m a t e

f o r I R ' ( x , k ) l g iven hereabove). Consequently (4 .2 .4 .2) h o l d s f o r k E Q+. However bo th s i d e s o f ( 4 . 2 . 4 . 2 ) a r e con t inuous i n k on @+\ {O} (see theorem 4.2.3 and i t s c o r o l l a r y ) . Hence w e can

-m -m X

X+m -m

conclude t h a t ( 4 . 2 . 4 . 2 ) h o l d s f o r a l l k E @ + \ { O } . 0

I n a d d i t i o n t o ( 4 . 2 . 4 . l ) w e s p e c i f y below t h e asymptot ic behaviour of R ( x , k ) f o r x -+ m and ~ ( x , k ) f o r x + -00, i f k E R and k + 0.

C o r o l l a r y I t o theorem 4 . 2 . 4 .

( 4 . 2 . 4 . 3 ) R (x ,k ) = r - ( k ) + r + ( k ) e 2ikx + o ( 1 )

L ( x , k ) = Il+(k) + !2-(k)e -2ikx + o ( 1 ) f o r x + -a.

f o r x + m

I n f a c t (4.2.4.3) i s o n l y a t r a n s c r i p t i o n o f a p a r t of t h e c o n t e n t s o f ( 4 . 1 . 1 2 ) i n terms of R and L . The o r d e r symbol o i s v a l i d uniformly i n k on compacta C R\{O 1 . I t is a n i c e and r a t h e r s imple excercise t o show t h a t t h e asymtotics g iven i n (4.2.4.3) combined wi th t h e i n t e g r a l e q u a t i o n s f o r R and L imply :

Lemma 4 . 2 . 4 . 2 . If u satisfies a W t h order growth condition,

then r + ( k ) and R-(k) for k E I R \ { O } are given b y :

( 4 . 2 . 4 . 4 ) r + ( k ) = !,e W

u ( y ) R ( y , k ) d y 1 -2 i k y

1 ," .2iky U ( Y ) L ( y r k ) d y L-(k) = -

2 i k -m


Let us conclude this section with an obvious consequence of

lemma 4.2,4,1 and part (a) of theorem 4.2.4.

Corollary 11 to theorem 4.2.4. If u s a t i s f i e s a 0 t h order

g r o w t h c o n d i t i o n , t h e n t h e a s y m p t o t i c s of r-(k) f o r Ikl + m ,

Im k > 0 i s g i v e n by m

r-(k) = 1 - - J u(y)dy + O(L) 1kI2

2ik -m (4.2.4.5)

4.2.5. The behaviour near k = 0.

If we suppose that the potential u satisfies a stronger growth-

condition, then it makes sense to consider R and L at k = 0

and these functions are quite regular there. The precise

formulation of these results is as follows :

Theorem 4.2.5. I. (a) L e t u s a t i s f y a I - s t order g r o w t h

c o n d i t i o n . Then t h e probZems(4.2.1.2,9 f o r R,L w i t h k = 0 are

u n i q u e l y solvabZe i n t h e s p a c e of c o n t i n u o u s f u n c t i o n s of x, w h i c h a r e bounded f o r x + - m , x + m.

R,L s a t i s f y (4.2.1.1). (i) ,(ii) i n cZass icaZ sense f o r aZZ

(x,k) E IR x $+. R,R',R",L,L',L'' a r e c o n t i n u o u s in (x,k) on IR x 2,. - (b) If u s a t i s f i e s a 2nd o r d e r g r o w t h condition t h e n aZso

~ , R L I R ~ , L ,L',L" a r e c o n t i n u o u s i n (x;k) on IR x z+ In the proof of the theorem we use the notation

x+ = max(O,x), x- = max(O,-x), x E IR.

k k k

We shall extend in an obvious way the definition of the rl - + +

operations * and * to functions E W and kernels S~;,where + + + + u

W i i I S ~ are found from P I S - by replacing the parameter space

&{Oi by $+. Using the following estimates for y G x, k E $+ :


we see that G' E So if [ 01 : G,G' E S o if [ 11 and Gk E So if

21, where kl is a shorthand notation for : u satisfies a

growth condition of order k.

k

Of courseranalogous statements hold for H,H',Hk,Hi.

It is further not difficult to verify that results analogous

to (4.2.2.3)and lemma 4.2.2.1 can be given with W-,S- replaced + +

Proof of theorem 4.2.5.1. ad a. The first part of (a) can

now be proven completely analogous to the proof of theorem

4.2.2. The crucial point is to find an estimate for the Gn's

valid also near k = 0.

It is a nice excercise to show that :

For the second part of (a) one proceeds as follows :

(i) one shows inductively : Gn E C ( I R

(4.2.5.2) implies R E C(IR x ?+) E C(IR x p+) and (iv) finally R" 6 C(IR x g + ) , because of (4.2.1.1). (i) . Further details are left to the reader.

x @+) (ii) next - (iii) hence R' = G' * R

ad b.

Now consider R

continuously extended from IR x f+ to IR x p+. equation (4.2.2.4Xi) : R = 1 + G * R we differentiate both

sides with respect to k. Using (4.2.3.4) we find on IR x $?+ :

(4.2.5.3)

This equation for Rk can uniquely be solved on IR E V+, i.e.

in Wo!

E Wan. We shall show that Rk can be

In the

- 1

R = G k * R + G * Rk. k

-

W e f i n d :

SCHRODINGER EQUATION

m

99

( 4 . 2 . 5 . 4 ) = x R

n > 0. %,n ' = G *

Rk,O = Gk * R ' R k , n + l

w i t h M(x) = su s u p I (Gk * R ) (y ,k) l . k$+ Y G

Next o n e shows i n d u c t i v e l y t h a t R E C ( I R x g+) . Hence b e c a u s e o f 4 .2 .5 .5 : Rk E C ( I R x F+).

D i f f e r e n t i a t i o n of t h e r e l a t i o n g i v e n i n ( , 4 . 2 . 2 . 9 ) : R ' = G * R

y i e l d s w i t h t h e u s e o f ( 4 . 2 . 3 . 4 ) on lR X @+ :

( 4 . 2 . 5 . 6 )

T h i s shows t h a t R i E Wan c a n be e x t e n d e d t o a n e l e m e n t

The proof o f (b) i s completed by u s i n g ( 4 . 2 . 1 . l ) ( i ) .

k , n

.. R i = G i * R + G' * Rk.

- -

E C ( I R x @+) .

A s for p a r t ( b ) of theorem 4 . 2 . 5 . 1 , w e h a v e t o require a 2nd order growth c o n d i t i o n f o r t h e p o t e n t i a l u. The m a i n r e a s o n t h a t w e have deduced t h e r e su l t s g i v e n i n ( b ) h e r e , i s t o show how a s t r o n g e r c o n d i t i o n o n t h e g r a w t h of t h e p o t e n t i a l i n f l u e n c e s t h e r e g u l a r y of t h e s o l u t i o n s a t k = 0 . A c o n s e - quence o f theorem 4.2.5.1-b i s t h e f o l l o w i n g e x t e n s i o n of t h e c o r o l l a r y t o theorem 4 .2 .3 , which p l a y s a n e s s e n t i a l ro le i n s e c t i o n 4 . 5 .

C o r o l l a r y t o theorem 4.2.5.1. If u satisfies a 2nd order

growth condition, then the transmission coefficient a is

c o n t i n u o u s at k = 0. r

Proof o f t h e c o r o l l a r y t o theorem 4 .2 .5 .1 Using t h e o r e m 4.2.5.1-b w e see t h a t t h e Wronskian of JIr a n d JIk : W(k) =

$,$; - $k$i = RL' - LR' + 2ikRL is a c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n o f k E p+. I n ( 4 . 1 . 1 6 ) w e found t h a t W(k) = 2 i k r - ( k ) . Hence w e c a n expand

100 W . ECKHAUS & A. VAN HARTEN

r - ( k ) near k = 0 i n t h e fo l lowing way : - r - ( k ) = - W(0) + & s ( 0 ) + w ( k ) 21k 2 1 dk -

with E C(g+)and l i m w(k ) = 0 . k+O

I m kZO

Because o f (4.1.15)we have f o r k E IR : I r - ( k ) I 2 1.

Th i s i m p l i e s :

(i) e i t h e r W ( 0 ) # 0

I n both cases ar = - is con t inuous a t k = 0. The d i f f e r e n c e

i s t h a t f o r k + 0 i n v+ w e have a r ( k ) + 0 i n case (i) and

a r ( k ) -* 2 i { ~ i i ; ( O ) }

It i s a l s o p o s s i b l e t o g i v e a r a t h e r s a t i s f a c t o r y e s t i m a t e o f t h e magnitude o f R , R ' , L , L ' v a l i d on a l l of IR X $+ and hence i n p a r t i c u l a r a l so n e a r k = 0.

d W (ii) o r x ( 0 ) f 0 , w ( 0 ) = 0

1 r-

0 dW -1 i n c a s e (ii).

-

Theorem 4 . 2 . 5 . 1 1 . If u s a t i s f i e s a g r o w t h c o n d i t i o n of o r d e r 1 ,

t h e n t h e r e i s a c o n s t a n t B > 0 o n l y d e p e n d e n t of u , s u c h t h a t

on ITI x @+ :

Proof of theorem 4.2.5.11. We s h a l l f i r s t demons t r a t e t h e estimate f o r [ R ( x , k ) I . Take xo E IR such t h a t

(., ~ U ( Y ) I ( l + ( y - x o ) ) d y f.

For x 2 xo w e have :

(4 .2 .5 .8) R ( x , k ) = g(x,k) + ( T R ) ( x , k ) def

x 2 ik (y -x0) w i t h g ( x , k ) = R 1 ( x o , k ) I e dy + R(xork )

xO


Then Vo is a Banach space with respect to 11 IIo.

Of course g is an element of Vo with a norm

T is a linear operator which waps Vo into Vo and the following argument shows that the norm of the operator T (i.e.

1 s u p IIT.Il /Ilvll is less than -- 0 0 2'

-0

X v+o

I (Tv) (x,k) 1 Q I l u ( y ) 1 (x-Y) Iv(y,k) Idy X

xO

Q Ilvllo(x-x0) I lu(y) 1 (l+(y-xo))dyG#llvllo(x-xo). since R = (~*)-'g w have xO

( 4 . 2 . 5 . 9 ) IIRIIO G 2 ltgll,

Using(4.2.5.2)it is easy to show, that:

sup sup IR(x,k) I exp(U1 (xo)+(xo)+Uo (x,,))

k€?+ x e 0

The relation R' = G'GR leads us to:

sup k€2+ x G o

sup IR'(x,k) I Q Uo(xo) exp(U,(xo)+(xo)+Uo(xo))

A combination of these estimates with(4.2.5.9 yields the

desired estimate for R.

The estimate for R' is an immediate consequence of the relation

R' = G'*R. O

We conclude this section with the remark that, if u satisfies

a growth condition of order 1, then the limits prescribed in

(4.2.1.1) for R,R',L,L' are uniform Qn p+. This improves the result given in the first part of theorem

4.2.4.b in this situation.


4.2.6. Parameter-dependent potentials.

Since we want to put a solution of the Korteweg-de Vries

equation (or some other time-dependent equation) as a potential

in the SchrGdinger equation, it is logical to consider at this

stage potentials u(xlt) depending on a parameter t E [TolT1].

Of course the functions R and L will then also be dependent on

this parameter t.

It is not difficult to obtain the following results:

Theorem 4.2.6. (a) Suppose: u E CClRx [TOIT1]),

max lu(xlt) I Q ;(XI and E s a t i s f i e s a growth c o n d i t i o n of tE[TO 1T11

o r d e r 0. Then t h e f u n c t i o n s R,R',R",L,L',L'' a r e c o n t i n u o u s i n

(xlklt) on I R x ( z + \ { O ) ) x [TolT2]. Moreover t h e s e f u n c t i o n s

a r e , f o r f i x e d x,tl a n a l y t i c i n k o n C+ and t h e i r d e r i v a t i v e s

w i t h r e s p e c t t o k a r e c o n t i n u o u s o n I R x @+ x [T ,T 1 . The a s y m p t o t i c s g i v e n i n theorem 4 . 2 . 4 a r e u n i f o r m i n

0 1

t E [TotTlI *

(b) Suppose , t h a t i n a d d i t i o n t o t h e c o n d i t i o n s i n ( a ) , :

. -

a growth c o n d i t i o n of o r d e r 0 . Then: Rt,R,'lR;ILt,L;IL; E

E C@R x ($+\EO)) x [Tot T 1. Moreover t h e s e f u n c t i o n s a r e ana-

l y t i c i n k o n C+ f o r f i x e d xI t and t h e i r d e r i v a t i v e s w i t h

r e s p e c t t o k a r e c o n t i n u o u s o n l R x $I!+ x [T o l T1l.

l

- Proof of theorem 4.2.6. (a) can be proven completely analogous

to the proofs of theorem 4.2.2.-3. The only difference is that

all functions depend on the parameter t. However the crucial

estimates (such as (4.2.2.7)) remain valid if we replace every-

where u by u. Gn is differentiable with respect to t with 5 E

E C@ E ($+\{O}) x [Tot T1]) and further that we have an esti-

(b) The idea i s to prove inductively that each

at


The estimate in (4.2.6.1) is obtained using the relation:

- aGn - _ - aGn+l - aG - * G n + G * = , n 2 0. It is now clear, that at at W

R = Z G is differentiablewith respect to t and Rt =

= Z

4.2.1.1-i one easily proves that also R+', R; E C ( R x ( g + \ { O ) ) x x [To, T1]). The proof is completed analogously to part a.

If in theorem 4 . 2 . 6 a,b we require 2nd order growth conditions

on u and Ll the results can be sharpened. It is then true that R,L and their derivatives

n wn=$G A

E C(IR x (g+\{O}) x [ T o , TI]). Using R' = G' * R!and at n=O

11+12+13

with l1 = 0, 1, 2; l2 = 0, 1; l3 = 0, 1 are a ax1' ak 2 at l3

elements of C W x ( ? + \ { O j ) x [T o r TII)*

In section 4.3 we shall show that these results imply cont.

differentiability of eigenfunctions corresponding to discrete

as well as non-discrete eigenvalues # 0, if u satisfies the

conditions in a and b.

d2

dx 4.3. THE SPECTRUM OF -7 + u ON L2(m) -

2 We consider the operator L with domain'HobR) = { J , E L2m) I + ' ' E L2W)] and LJ, = - + ' I + uJ,. Note that J, E H O W ) * L$ E L2W) because of the boundedness of the potential u, see (4.2-3).

L is of course an unbounded operator on the Hilbert-space

L2(IR). Furthermore L has some nice properties, which are

important for the spectral analysis. We shall show that L is

closed and symmetric (for definitions of these concepts, see

Kato, 1966).

Let <,> and I I 11 be the usual innerproduct and norm on L2 m). Considering the closedness we reason as follows. Suppose

Qn E H O W ) for n E N and J,n ny J, in L2W), Lqn np 0 in

L2(IR). Then is a convergent sequence in L2W) with limit

u$ - 4 . But in distributional sense it holds that J,: "=o" + ' I . Using the uniqueness of distributional limits we

2

2


see that JI" E L2 (R 1 and JI" = uJI - 0. Hence, indeed : Q E HO(R ) and Lg = 4 . 2

The symmetry of L means that

(4.3.1) (LJI ,$) =($,LO) V$,Q E HO(R 1 . 2

This is easily verified by integrating by parts twice.

The SchrGdinger equation given in 4.1 has a clear connection

with the spectral equation

(4.3.2) (L-A) Q = 0.

As usual for a closed operator (see Yosida, 1974, pg. 209) we

define the resolvent set p ( L ) as the subset of g! consisting of

those A's for which L-X is injective and surjective and has a

bounded inverse.

Hence X E p ( L ) means that the problem

(4.3.3) (L-X)Q = f

2 possesses a unique solution JI E HO(R ) for each given f E L2(R)

which satisfies an estimate

where the constant C ( h ) is independent of f.

Theorem 1, pg. 211 in Yosida, 1974 shows that p(L) is an open

subset on Q1 . The complement of the resolvent set is called the spectrum of

L. We shall denote the spectrum of L by u ( L ) . A sufficient condition in order to have X E u(L) is that there exists a sequence {Qn;n E N)

n E N and

2 in HO(R ) with IIQn! # 0 for all

For self-adjoint operators this condition is also necessary

(see Yosida, 1974, pg.319).

A special case for this condition arises when X E a ( L ) is an

eigenvalue, i .e. when the equation (4.3.2) for this X has a


2 non-trivial solution + E Ho ( R ) . An eigenvalue A E a(L) is called isolated if distance 0 (AO,a(L)\CAOl) > 0. We define IPo(L) as the subset of a(L), which consists of all isolated eigenvalues. Because of the

symmetry of L all eigenvalues are real.

In the following theorem we express p(L), U(L) and IPa (L) in

terms of the scattering coefficient r-.

Theorem 4.3.1. L e t t h e p o t e n t i a l u s a t i s f y a 0 - t h o r d e r growth

c o n d i t i o n . Then : n

(i)

(ii)

p ( ~ ) = { A E @ l A = kL w i t h k E 2+ s u c h t h a t r-(k) # 0 )

IPa(L) = { A E Q l A = k 2 w i t h k E @+ s u c h t h a f r - ( k ) = O ) ,

IPO(L) c (-R,o) IR for some R > o (iii) u ( L ) = IPu(L) U [ O,m) C IR ,

(0,m) does not contain eigenvalues.

The proof of this theorem will be completely based on the

properties of the solutions JI, and JIr of the Schrbdinger

equation introduced before.

We shall split up the proof into a nurnber of steps (a),(b)

and (c).

In this situation JIQ(x,k and $ (x,k) are two linearly

independent solutions of .the Schrbdinger equation, for

JIR(x,k) grows/c!ecays exponentially for x -t -m/x + +m and

+,(x,k) grows/decays exponentially for x * +m/x * - w , see

(4.2.4.1). Hence there are no solutions of the equation

(L-A)JI = 0 in HO(R).

r

2

We define a Greens kernel for L-h in the following way :

'D (k) L (5 ,k) R (x,k) e-ik(x-5) I S 2 5

I X 2 5 ik (x-6) c D(k)R(E,k)L (x,k) e (4.3.6) Gr(x,[,k) =

with D(k) = [ 2ik r- (k) I - l .


Using (4.2.2.5-7-8) we see that Gr (x, 6 I k) satisfies the

estimate :

(4.3.7) IGr(x,S,k) I Q A(k) exp(-Imk. Ix-SI)

with A ( k ) = ID(k) I exp ( 2 J lu(y) ldy/lkl).

It is easy to verify that the equation (L-A)$ = f (i.e. 4.3.3)

with f E D = {h E C(R 1 I support (h) is compact) has the

solution :

(4.3.8) Jl(x,k) =-I Gr(x,S,k)f(S)dC . This solution satisfies the estimate':

m

-m

0

m

-m

with a certain constant C(k) > 0. This estimate is not trivial and the derivation is given

2 below. In the calculations we put B = A(k) ,a

-a(lx-S1l+lx-S21) lJl(x,k)I2 Q B I e I f (5,) I

-m -m

O0 -a(lx-5J+lx-S21) -0. I 5,-S2 Using that J e dx = e

we find :

-m

1 +;I

By Schwartz inequality we have : m

lf(C+n)f(C-n) ldrl Ilf1I2. -m

Hence :

This proves (4.3.9)


I t is now c l e a r t h a t $ E H;(W 1 . Fur thermore JI is t h e unique 2 s o l u t i o n o f (L-A)$ = f i n H o ( W ) , s i n c e t h e homogeneous

2 e q u a t i o n has no s o l u t i o n s i n HO(m 1 .

Using t h e f a c t t h a t D o i s dense i n L 2 ( W ) w e can ex tend t h e s e r e s u l t s t o a l l f E L 2 ( 1 R ) . Because of (4 .3 .3 -4 ) th i s y i e l d s :

Lemma 4.3.1. r-(k) f 01

(b) : h = k wi th k E &+ such t h a t r - ( k ) = 0 .

p ( L ) 3 {A E C I A = k 2 w i b h k E $?+ s u c h t h a t

2

I n t h i s s i t u a t i o n $r i s p r o p o r t i o n a l t o $

i n t h e f o l l o w i n g way. L e t xo E IR b e such t h a t IL (x ,k ) I 2 4 f o r x 2 xo . Then J I I I (x lk ) and G R ( x I k ) = J I I I (x lk)

are l i n e a r l y independent s o l u t i o n s of t h e Schrod inge r e q u a t i o n f o r x 2 xo . Hence JIr!x,k) = a J I R ( x I k ) + f3JIR(xIk) f o r x 2 x w i t h a,B E $?.

I f B # 0 t h e n l $ r ( x l k ) I 2 C exp(1m k .x ) f o r x + m w i t h some c o n s t a n t c > 0.

However [ Q r ( x I k ) I = IR(x ,k) lexp(1mk.x) and because o f (4.2.4.1): l i m l R ( x , k ) I = 0 .

T h i s g i v e s a c o n t r a d i c t i o n . Hence J I r (x ,k) = aJ IR(x ,k ) fo r x > x b u t t h e n t h i s i s n e c e s s a r i l y t r u e o n a l l of XI.

T h i s c a n be s e e n

IJI,(E,k)-2dE

II'

X

xO -

0

X'm

0 '

W e conclude t h a t JIr is a s o l u t i o n of (L-A)$ = 0 which d e c r e a s e s e x p o n e n t i a l l y t o bo th s i d e s , i . e . A is an e i g e n v a l u e and JIr a n e i g e n f u n c t i o n E Ho (R 1 . T h i s i m p l i e s t h a t h is real i . e . k = i p w i t h 1-1 E R , u > 0.

Consequent ly :

2

(4.3.10) h = - ) J L < 0.

Using GI.2.2.5-7)we f i n d :

m

w i t h uo = I u ( y ) d y . -m


In combination with(4.2.4.1) this estimate shows that 2 K > 0 such that

L Hence if we put = -K , then :

(4. 3.12) A >-Q.

Finally we use that r- is analytic on $?+, 1.e. the zero's of

r- in @+ are separated.

Because of lemma 4.3.1 we can conclude :

2. Lemma 4.3.2.

r-(k) = 01. ? . { A E @ \ A = k C ( - Q , O ) C IR.

- 1. I P o ( L ) > { X E $ I X = k 2

w i t h k E @+ s u c h t h a t

w i t h k E $+ s u c h t h a t r - ( k ) = 0)

2 (c) : A = k with k E IR, k # 0, i.e. X E ( 0 , m ) C m .

In this case $r and 5, are two linearly independent solutions of (L-A)$ = 0. These solutions behave oscillatory for 1x1 -+ 00.

Hence X cannot be an eigenvalue. However, the following

reasoning shows that nevertheless A E o ( L ) .

Let x be a cut-off function E C"(IR) with x(x) = 1 for 1x1 < 1, x(x) = o for x 2 2, Ix(x) I G 1 on IR.

Define xn by xn(x) = x (F).

Consider the sequence $n = Qrxn, n E N.

It is clear that :

X

(4.3.13) IIQn l l .f for n + m.

An easy calculation shows that :

Now we have :

IJlrx; + W'X'I = 0 outside [n,2nl U [-2n,-nl

hrx; + W'X'I G - C (k) on [n,2nl U [-2n,-nl

r n

r n n


with a certain constant C(k).

This leads to :

.. -n 2n

Consequently :

Because of (4.3.5) we have X E u (L) ! Hence :

Lemma 4.3.3. u ( L ) 2 lR+. IR+ does n o t conta in e igenva tues .

The proof of the theorem is now completed by noticing that

0 E u ( L ) since a ( L ) is closed. Of course 0 f IPa(L) for 0 is a non-isolated point of a ( L ) . 0

Theorem 4.3.1 shows that the X's in the spectrum u ( L ) coincide

nicely with physically interesting values of the spectral

parameter as indicated in the introduction of this chapter.

A consequence of theorem 4.3.1 is that 'the operator L is self-

adjoint, seeKato, 1966, theorem 3.16, pg.271.

Of course it'is also possible to.see that L is self-adjoint

by different methods,for example using the theory given in

Kato chapter V, 15.2 and 94.4, theore'm 4.3.

Because of the selfadjointness of L it holds that

(0,m) C a ( L ) consists of the so-called continuous spectrum, see

Yosida, XI. 8, theorem 1.

Till now it is not clear whether = 0 E u ( L ) i s an eigenvalue or not. If u satisfies a growth condition of order 1 it is

easy to verify using the results of 94.2.5 that O E u ( L ) cannot

be an eigenvalue.

Under this condition a l l of [O,m) consists of continuous


spectrum.

A useful supplement to theorem 4.3.1 is:

Corollary 4.3.1. Each e i g e n v a l u e A = k E IPa(L) i s s impZe. The

one-dimensiona2 e i g e n s p a c e E ( X ) i s spanned b y t h e r e a l f u n c t i o n

JIr(-?k). Furthermore JIr(-,k) = a(k)JIQ(-?k) with

a ( k ) # 0 .

2

a(k) Em,

The proof is elementary and is left to the reader.

A deeper result is the following one.

Theorem 4.3.11. If u s a t i s f i e s a 1 s t o r d e r growth c o n d i t i o n ,

t h e n t h e number of d i s c r e t e e i g e n v a l u e s i s J ' i n i t e { i . e . IPa(L)

of d i s c r e t e e i g e n v a l u e s N > 0 i s a f i n i t e s e t } . The number

s a t i s f i e s t h e e s t i m a t e

(4.3.15). N 2 + .f I y I l u ( y ) m

-m

Proof of theorem 4.3.11. The proof of this result is based on

a so-called comparison theorem, which we give in (i). In (ii)

we apply this comparison theorem to the eigenfunctions corres-

ponding to discrete eigenvalues. Finally in (iii) we shall

show that (4.3.15) has to hold . (i)

We consider two real classical non-trivial solutions JIo and J12 of the SchrGdinger equation with the spectral parameter equal

to X o E Q? and A 2 E IR with A. *< A 2 g 0 (i.e. JIi; + (Xo-u)JIo= 0, JI" 2 + (A2-u)IJ2 = 0).

Let a and b be consecutive zero's of JIo with a < b. We want to allow a = -m and b = + m . Of course we shall call - m l +m a zero

of q0 if lim Q0(x) = 0 or lim JIo(x) = 0.

Now the following comparison result is valid

XS-m X + m

3c E(a,b) such that J12(c) = 0.


In the case a ElR, b EIR this result follows from theorem 1.1,

ch. 8, Coddington, Levinson, 1955.

If a ElR and b = the reasoning proceeds as follows.

First we derive estimates for $

$ o = const - $ R ( - , i m ) it is clear, that $

exponentially for x + a . Using the SchrGdinger equation and an

interpolation argument we find that also $I' and $ ' decrease

exponentially for x -+ m. The condition in (4.3.16)implies

I$,(x) I < C(l+lxl) for x -+ a. Hence, using again .the

Schrgdinger equation and an interpolation argument, it is clear

that also $i and $ > grow at most linear in absolute value for

x -+ m.

As a consequence products such as $, (x)$' (x) and $ (XI$; (x)

tend to 0 for x -+ =.

and JI , $ I for x -+ a. Since 0' % 2 2 decreases . a

0 0

0 0

Let us now suppose that I$,(x) I > O on (a,m). It is no

restriction to take $,(x) > 0, $ (x) > 0 on (arm). An easy calculation shows that :

0

m

O = I [i$i+(AQ-~)$ol$2 - {$;+(A~-u)$ I $ I (x)dx 2 0 a

m m

This is however a contradiction for $,(a) > 0 and $' (a) 2 01

The conclusion is that J12(x) has a zero somewhere in (a,=).

The other cases : a = -00, b < m and a = - 0 0 , b = m are left to

the reader as exercises.

0

(ii)

It is already known that the discrete eigenvalues can be given

as a sequence : -a < -ul < -p2 < .... < -p <.. ...... < 0. Let us denote a real eigenfunction corresponding to the

discrete eigenvalue -pn by $n. Repeated application of the

comparisofl theorem given in (i) shows that $n has to have at

least n-1 zero's in R besides of its zero's in --m and +m .

n


(iii)

Now consider the real solution J, = R(X,O) of the Schrbdinger

equation with X = 0. We shall demonstrate that the number of

zero's of J, in R can be estimated by

(4.3.17) )# tx E R $(XI = 0) 2 + I IyI lu(y) Idy.

This estimate will be derived analogously to exercise 3, pg.

255, Coddington, Levinson, 1955. Suppose that J, has consecutive

zero's a,B E R . Let $ be the solution of $" = -lul$, $ ( a ) = 0

$ ' ( a ) = $ ( a ) . If u Q 0 on (a ,B) then $ = J, on [a,BI, i.e.

m

-m

$ ( P I = 0 . If u > 0 somewhere in (a,B) then, using a convexity

argument, it is easy to see that $ ( y ) = 0 for some y E (a,@). z -

Hence, there exists a first point y E ( a , P ] , such that

$ (y) = 0. For x E [ a,yl we have the identity :

X $(XI + I (x-S) ]U(S) I$(s)ds = $ ' ( a ) (x-a).

a

I$(x)I I ~ , ' ( a ) l (x-a) on [a,yl Consequently :

and

0 = l $ ( Y )

l $ ' ( a

= IJ,

(r-a

In this way we obtain :

Y 1 4 I ( Y - s ) ~u(s) Ids

CL

Y 1 4 I ( s - a ) ~ u ( s ) Ids.

a

This leads us to the inequality :

B

a 1 4 / I s 1 lu(s)lds if a > 0 or B Q 0.

It is now immediately clear that : m

)# {x E [O,-) [ $ ( X I = 0) 1 + IlYl lU(Y) IdY 0 0

x tx E (--I01 l$(X) = 0) 4 1 + J A Y 1 lu(y) IdY.


Herewith (4.3.17)has been derived.

Suppose that the number of discrete eigenvalues exceeds N with

N > 1 t 1 lyl lu(y) lay. The proof is then easily completed by deriving a contradiction. We saw in (ii) that $N+l has N zero's

in R . Because of theorem 4.2.5.11 we can apply the comparison

argument given in (i) with J, - and $, = JI . The conclusion 1 'N+1 would be that J, has N+l > 2 t 1 IyI lu(y) (dy in R in contradiction with c4.3.171 0

m

-m

-OD

Next we shall derive a formula for the derivative of r- in a point k E g+ where r-(k) = 0 . In particular this formula will

show that the zero's of r- are of 1st order.

Lemma 4.3.4. S u p p o s e X = k E IP u ( L ) . Define a(k) E R I

a(k) # 0 QS in corollary 4.3.1. T h e n

2

d*- 1 2 (4.3.18) - = - dk ia(k) iJIr(*'k)'

Proof of lemma 4.3.4. In section 4.2.3 and 4.2.4.1 we have

found that R(x,k) is a family of analytic functions of K E g+ parametrized with xI which converges far x- uniformly on wnpacta

C gt to the analytic function r-(k). An application of a well-

known theorem given for example in ConwayI 1973,th.2.11 pg.147

we find that :

yields :

Analogously

lim %(xIk) = lim $(x,k) = 0

lim %(x,k) = 0 .

x+-m x+-m

X+=

The trick is now to consider the following function :

(4.3.19) w = (Ri R-%R') e-2ikx.

Using the differential equations which R and \ have to satisfy (R" = 2ikR' + uR ; (see (4.2.1.1)(i)) and the limits for x -* -0 of RIR'I\ and

(4.2.4.1) and here above) we find the following problem for UJ :

= 2ikS + u% + 2iR' (see

114 W. ECKHAUS 5 A. VAN HARTEN

-2ikx F w ’ = 21 R’R e

= i( (Qr) 2 ’ + 2ik

lim u(x,k) = 0. c X+-m The solution of this problem is

2 (4.3.20) u(xrk) = i+r(Xrk) -

Next we use the fact that kL E IPa(L) . From the relation + r = ct(k)+& we deduce that for such k :

-2ikx - lim R(x,k)e - a(k)

lim R’(x,k) e -2ikx = 2ika(k). X’-

X’m

A calculation of lim u(x,k) for k‘ E IP a ( L ) in both (4.3.19) and (4.3.20) leads us to :

X’m

dr- m 2 - Ziku(k) K ( k ) = -2k J +,(Yrk)dy. -m

This proves 4.3.18 since JIr is real if kL E IP o(L) . The lemma shows that the poles of the meromorphic function

a = r-

enables us to calculate the residuals of ar in its poles. We

shall use these facts in section 4.5.

-1 on fZ!+ are of first order. The formula (4.3.18) r

We now consider potentials u(x,t) depending on a parameter

t E “J?orTII . The t-dependence of the operator L will be expressed by the

notation L(t).

It is easy to imagine that discrete eigenvalues

X(t) E IPa(L(t)) can form nice trajectories when t varies.

We shall call lX(t) ; t E I} a C -eigenvalue trajectory with

existence interval I c [ T o r T 1 ] , if X : I + (-m,O) is

continuously differentiable, X(t) E IPa(L(t)) for all t E I

and if the function X cannot be extended to a larger interval CITorT1] with preservation of these properties.

1


We allow the interval I to be open, closed at one end or closed

i.e. I has the form (tO,tl), [tortl),

With these preliminaries we formulate :

(tortl] or [tortl].

Theorem 4.3.111. Suppose t h a t t h e p o t e n t i a l s a t i s f i e s t h e

c o n d i t i o n s g i v e n i n t h e o r e m 4 . 2 . 6 ( a ) and ( b ) . Then :

( a ) :

( i ) -

g i v e n t

a u n i q u e C 1 - e i g e n v a l u e t r a j e c t o r y (X (t);t E I 1

E [To,T1] and X o E IPa(L(tO)) t h e r e e x i s t s 0

w i t h

0' X(tO) = x ( i i ) e i g e n v a l u e t r a j e c t o r i e s c a n n o t i n t e r s e c t .

( i i i) i f t h e e x i s t e n c e i n t e r v a l of a t r a j e c t o r y i s o p e n a t

t+T tEI

t h e e n d p o i n t T, t h e n lim A(t) = 0

( b ) :

f o r each e i g e n v a l u e t r a j e c t o r y {X(t) ; t E I ) t h e c o r r e s p o n d i n g

e i g e n f u n c t i o n s $r (x,k(t) ,t) w i t h k(t) = i d m h a u e

$r d e r i v a t i v e s ~

t h e norm I[$.(. ,k(t) ,t) 11 on I .

-

an tX,

E C ( m x I) ; n = 0,1,2; X = 0 , l and axnatx

i s c o n t i n u o u s l y d i f f e r e n t i a b l e i n t

( e l :

g i v e n X = k E (O,m),kEIR f i x e d ( i . e X is a f i x e d p o i n t # 0

i n t h e c o n t i n u o u s s p e c t r u m ) t h e g e n e r a l i z e d e i g e n f u n c t i o n s

2 -

an+4 a n t x ~ II

ax at axndtx $r(x,krt) ,$,(Xrkrt) have d e r i v a t i v e s - nx JIr , - EC(Rx[TOrT]);

n = 0,1,2, Y = 0,1.

Proof of theorem 4.3.111. Part (c ) of this theorem has already

been proved in theorem 4 . 2 . 6 .

As for the proof of (a) and (b) the reasoning is as follows.

As a consequence of theorem 4 .2 .6 we find that r-(k,t) is a

E C y ($+ x IR ) by putting :

C'-function of (k,t) on (?!+ x [T orTII

We extend r- to an element


r-(k,t) = r-(k,t) t E [To”l’lI

dr- r-(k,Tl) + (t-T1) =(kIT1) t > T1 . I

Now suppose that to E [TO,T1l, X o E IPa(L(tO)).

Then with ko=i we have:

r,(koItO) = 0

where the latter inequality follows from lemma 4.3.4, 4.3.18.

Hence we are precisely in the situation where the implicit

function theorem is applicable (see I Rudin, 1964).

This yields that there is an

the equation r-(k,t) = 0 has a unique solution k(t) in

C [ - E , E ] , which depends continuously differentiable on t on

Of course a-(i) follows now almost immediately.

E > 0 such that for (tO-tl G E

[ - € , E l .

As a consequence of theorem 4.2.6 the functions $r(xik(t)it)

and $,(x,k(t) ,t) have derivatives upto the order 2 in x and

1 in t, which are elements of C ( I R x I).

This proves the first part of (b).

Using (4.3.18) we find :

Jlr(x,k(t) ,t) ar - II, (x,k ( l+r(.l k(t),t)I2 = i t) ,ti (k(t) it)

with x E R sufficiently large to ensure $E (x,k(t) It) # 0 on

“I‘OtTII - From this expression the contents of the second part of (b)

can easily be derived.

Now suppose that two or more trajectories intersect at t = to

in X o E IPo(L(tO)). The theory given in Kato,l966,ch.VI 14.3

shows that X o would be a multiple eigenvalue of L(tO). This contradicts corrollary 4.3.1 and the conclusion is that a-ii

holds.


As for a-ifiwe note that A(t) E (-a,O) with Rindependent of

t E [ T ~ , T ~ I , see 4.3.11-12 . This implies that either 1. lim X (t) = 0 br

t+T tEI

2.

such that lim X(tn) = X o .

Suppose that the statement 2. were true.

The thaq given in Kato,1966, ch.V, 14.3 shows that X o E IPa(L(T)). Let {T(t),t E ?I denote the eigenvalue trajectory trough

lo < 0 and 3 a sequence {tn;n E IN 1 with tn E I, lim tn=T n

n+m

X o E IPa (L(T) 1. Since I n f contains an interval with endpoint T and trajectories cannot intersect we must have lim h(t) = X 0' t+T

tEI

Then,because of the uniqueness of the trajectory through

X o E IPU(L(T)) we have X(t) = T(t) on I. The original trajectory

can be extended to I U ?, in contradiction with the definition

of I. 0

Note that eigenvalue trajectories either exist on all of

[To,T1] br they start or end in 0 E &.

The contents of the theorem show that the differentiations

with respect to t in chapter 2 (such as in (2.2.4), (2.3. 1.3))

are allowed.

It is left as an exercise to the reader to investigate how

theorem 4.3.111 changes, if we only impose the condition

on the potential u given in theorem 4.2.6.a.

4.4. FOURIER TRANSFORM OF THE SOLUTIONS.

One of the strongest techniques to obtain information about

solutions of linear differential equations is Fourier

transform. In our situation where functions such as R,L,+r and

+g depend on x and k we shall apply Fourier transform with

respect to the second variable k.

Here below we shall first give a brief survey of the usual

Fourier theory. Next we consider Fourier transform with

respect to the second variable k. The theory for that


transform is quite analogous to the usual one. In fact the

first variable x acts only as a parameter. However, we deal

rather explicitly with the second case in order to specify

continuity results with respect to the first variable. After

that we shall derive a number of properties of the Fourier

transforms w.r.t. the second variable of L and $L. These

properties will be used in the next section.

For the usual theory of Fourier transform we refer to Yosida,

1 9 7 4 ; Hgrmander, 1 9 6 3 ; Rudin, 1 9 7 3 ; Schwartz, 1 9 6 5 and Gel’fand, Shilov, 1 9 6 4 . In this theory one defines Fourier transform F and inverse Fourier transform F - l first on the Schwartz space 6 , which consists of rapidly decreasing Cm-functions. In the one-

dimensional case, = Q (IR ) , this is done as follows : m

( 4 . 4 . 1 ) ( F $ ) ( s ) = (27r)-’ J $(k)e-iksdk, vs E IR -a

m

( F - l @ ) (k) = (2n)-’ $ ( s ) eiksds, Vk E IR

The linear operators F and F-’ are 1-1 fm G onto 6

satisfy F F - ~ = F - ~ F = 1.

Moreover these operators are continuous with respect to the

usual topology on 6.

It is also possible to extend F and F - l to 6’, the space of

tempered distributions. This is done by the method of

dualization. Let ( x ,$ ) denote the action of an element

x E 6’ (i.e. a continuous linear functional on 6) on a test- function $ € 6. On 6’ we define F,F-’ by :

-a

and they

( 4 . 4 . 2 ) ( F x , $ ) = ( x , F @ ) V@ E G

( F - ’ x , $ ) = (x ,F- ’@) v@ E G

In this way F and F- l become injective linear operators from

$’ onto Q ’ , which coincide on ti5 with the previously defined

operators and which are inverses to each other: F F - l = F - l F = l . These operators F and F - l are continuous with respect to the

weak topology on 6’.

SCHRbDINGER EQUATION 1 1 9

It is also useful to know how F and F- l act on L1 and

L2 C (3' (L1 = L1 (IR) , L2 = L2 and F - l is still given by (4 .4 .1 ) (the integrals in the right- hand side converge absolutely). Let us define

Co = {u E C(1R)Ilim Iu (k ) I = 01. We equip Co with the

maximum norm and L~ with its usual norm. Then F and F continuous from L1 into Co. On L 2 the situation for Fourier- and inverse Fourier transform is even more beuatiful. F and F - l are not only continuous from L2 onto L2 : they preserve the L2-norm.

(R ) ) . On Ll the action of F

Ik l - -1 are

-

Since L2 n L1 is dense in L 2 we can find F$ as lim FOn in L2 n+- ~~

with 0, E L 2 n L1, +n -+ 0 in L2 and F$n given by 4 . 4 . 1 . Of

course the action of F - l on L2 can be described analogously.

We shall now deal with Fourier and inverse Fourier transform

w.r.t. the second variable for certain functions and

distributions depending parametrically on a firts variable

x E l R .

Our notation will be as follows. V will denote a linear

topological space, such as GIG' ,L1,Lz or Co. F(R -+ V ) is the set of functions on R with values in V. The value of

I$ E F(IR -+ V) in x E IR is denoted by $ ( x , . 1.

We def

F(IR -f

( 4 . 4 . 3

-1 ne linear operators F2 and F2 a ' ) by :

from F(R -+ C ' ) into

Of course F2 is the Fourier transform operator w.r.t. the

second variable and F i l is the inverse Fourier transform w.r..t. the second variable. Because of the properties of F and F - l it is clear that F 2 and F;' are injective from

F(R + a ' ) onto F(IR -+ a ' ) and that F 2 F i 1 = F F = 1.

Moreover F2 and F i l have the following behaviour on subspaces of F(R -f a ' ) : they map F(lR -+ a) onto F(R + G I , F(R + L 2 ) onto F(R + L 2 ) and F(R + L1) into F(R + Co).

However, more interesting is what these operators F2 and 'FZ1

-1 2 2


do with functions and distributions, which are continuous in x.

Let us define : C ( I R -+ V) = CQ E F(IR + V) IQ is continuous}.

Using the continuity properties of F and F-' it is easy to

verify that

(4.4.4) F2 and F2 map : C(IR + a ' ) onto C ( I R + a ' ) , C ( I R + G ) onto C(IR -+ G ) ,

C ( I R * L2) onto C(IR -+ L2),

C ( I R -+ L1) into C ( I R -+ C o ) .

-1

In fact it can even be shown that F2 and Fil are continuous as

linear operators between the spaces mentioned in(6.4.41, if

these spaces are equipped with the "Socally uniform in x" topology, but we shall not need this fact.

Let us next consider the function L(x,k) as defined in

8.4.1.1) (ii). We shall assume that the potential u satisfies

a growth condition of order 1. Then L is for each x E IR,

continuous in k E IR (see theorem 4.2.5.1) and bounded

uniformly in k E IR (see (4.2.5.7)). As a consequence we can

interprete L as an element of F(IR -+ a ' ) and apply Fourier transformation w.r.t. the second variable to it. Let us

define :

i.e. L = 1 + JT~~.F;~J.

It is easy to explain why we prefer to work with L-1 rather

th.n with L itself.

Using theorem 4.2.4 we see that IL(x,k)-lI = O(]kl'l) for

Ikl -+ m uniformly in x E IR . Hence for a fixed x E IR

function L(x,.)-1 is in L2, i.e. L-1 E F(IR + L 2 ) . Because of the continuity of L in x and k on lR x IR (see theorem 4.2.5.11

one finds that it is even true that L-1 E C(IR + L2). This

property of L-1 implies that J as defined in (4.4.5) is an

element of C(IR -+ L2) (see (4.4.4)). However a lot more can be

derived about the properties of J.

the


Theorem 4.4. Suppose t h a t t h e p o t e n t i a l u s a t i s f i e s a g r o w t h

c o n d i t i o n of o r d e r 1.

Then J = (2v)-’F2(L-1) i s a n e l e m e n t of C(IR -+ L2) n C ( I R -+ L1)

w h i c h can be i d e n t i f i e d w i t h a f u n c t i o n :

(4.4.6) J(x,s) = N(x,s) for s 2 0

0 for s < 0

As a c o n s e q u e n c e t h e f o l l o w i n g F o u r i e r i n t e g r a l e x p r e s s i o n

h o l d s :

(4.4.7)

The k e r n e l N i s an e l e m e n t of Co(x:) =

= Iw E C(IR x [ O l m ) ) IVx E IR : lim w(xIs) = 0 1 , w h i c h i s

d i f f e r e n t i a b l e w i t h r e s p e c t t o x a s w e l l a8 t o s and

m

L(x,k) = 1 + S N(x,s) eiksds. 0

s+m

F u r t h e r t h e k e r n e l N i s r e a l and has t h e i m p o r t a n t p r o p e r t y

(4.4.8) N(x,O) = f I u(y)dy, vx E IR

i . e .

m

X

u(X) = -2 Nx(x,O) vx E IR

Proof of theorem 4.4. The proof is set up in the

following way. We start with the deduction of a nice

representation of J, from which we at once find (4.4. 4, (4.4.8)and the fact that N E Co(x;). Next we investigate the

derivatives Jx and Js and we show that Nx,Ns E Co (?ir:). Finally we derive a first order integro-differential equation

for N which we use in order to demonstrate that N(x,.)

E L1(O,m), Vx E R . This implies (4.4.7) where the

integral in the right-hand side converges absolutely.

(i 1 First we shall prove that J can be given as follows


with: H(s) =

co

U ( z ) = I u(y)dy

J E c (IR -+ co), ~ ( x , s ) = o for s

Z - 0.

This is a consequence of the expansion given in theorem 4.2.2: W

L - l = H + zHn. n=2

Here H1 is given by:

m m

dy = # I H(s)U(x+fs)eiksds. 2ik (y-x) = I U(y)e

X -00

m

It is easy to check that IHl(x,k) I Q I lu(y) I (y-x)dy and that IHl(x,k) I = O(1kl-l) for k -+ m uniformly in x. Further H1 is

continuous in x and k on IR x IR . We conclude, that H1 E C(IR -+ L2) and (4.4.4)then implies F2Hl E C(IR -+ L2).

It is not difficult to calculate F 2 ~ 1 explicit1y.m easy way isb note that vx E R : H1(X, . I = k$-lti(x, . I with ~(x,s) = H(s)U(x++). m e

imnediate conclusion is that H (x, . = 4 u (x, . ) Let us naw first give a useful lemna.

X

V

1

Lemma 4.4.1. Let $I be a coxtinuous function of k E c, which is anaZytic on C+. Suppose, that $I restricted to the

real axis is in L

I IR

and that sup- Ik$(k)I < m. Then: k E C+ 1 -

$I(k)e iksdk = 0 for aZZ s E IR with s < 0.

Proof of lemma 4.4.1. Let A be the contour consisting of

[-R,R] C IR and AR = (k E C+IIkl = R). Of course

J

contour and continuous upto the contour.

The result of the lemma is found by taking the limit

R + a. It is left to the reader to show, that

I $(k)eiksdk + 0 for R -+ m, if s E IR and s < 0.

R

$ (k)eiksdk = 0 for the integrand is analytic inside the AR

a, 0

SCHRODINQER EQUATXON 123

m As for I: H we note, that this function equals

L-l-H1 and is therefore, continuous in x and k on IR x IR (see theorem 4.2.5.1) and for a given

x E IR for all k E IR bounded by l+B(l+x-)

+ J lu(y) I (y-x)dy (see (4.2.5.7) and the estimate for H1). Moreover 1 2 Hn(x,k) I GGlkl = O(lkl)-2) for Ikl + m (see theorem 4.2.1).

The conclusion is, that

We define J = (2n)-’F2( I: H ) . Then because of (4.4.4)

n=2 n

m

X -2 m exp(1kl-l /” (U(Y) ldy)

n= 2 -m

Hn E C(IR + L1). g= 2 - n=2 n : E C(IR + C0).

Using theorem 4.2.3 and theorem 4.2.5.1 we find that

I: H n=2 n for each x E IR. Since the other conditions of lemma

4.4.1 have also been fulfilled we have:

m is analytic in k on C, and continuous in k on F+

J(x,s) = 0 vs E IR-.

- - The continuity of J implies then that J(x,O) = 0.

Herewith the proof of (4.4.9)is complete.

It has now been shown, that(4.4.6)holds with N E Co(lR+) T and N(x,s) = #U(x+$s)+;(x,s).

The contents of CQ.4.8)have also been verified, since

J(x,O) = 0.

It is left as an exercise to the reader to show, that

L(x,-k) = L(x,k) for all k E IR and to demonstrate,

that this implies that N is real.

-

(ii)

Here we shall demonstrate that it makes sense to

differentiate J with respect to x and s and we shall

specify some properties of these derivatives J and Js. X


the Taylor expansion L(x+h,k) = L(x,k) + hL' (x,k) L" (x+<,k) (h-<f' dc and the estimate +

max ?L"(y,k) I (combine (4.2.1.l)dii) and (4.2.5.7) ) we see that for

each x E lR:

B(21kl + max lu(y) I (l+y-)), I C IR compact Y E 1 Y E 1

lim h-l{(L(x+h,*)-l) - (L(x,-)-l)} = L' (x,.) in h+O hfO

Operating on both sides of this equality with (2.rr)-'F

yields that J(x,-) is differentiable in each point

x E IR and that Jx(x,-) = (21r)-'FLl (x,.). Since L' is

continuous in x and k on IR x IR (see theorem 4.2.5.1)

and bounded uniformly by a constant B (see 4.2.5.7)

we have L' E C(IR + G I ) . Consequently:

Jx = (27r)-'F2L' E C(IR -* G I )

Next we use the formula (4.2.2.10t m

V V L' = Hi * 1 + HI * H1 + 2 HI Hn n= 2

where :

Analogously one shows, that:


We conclude that Jx E C ( I R -+ L2) and

(4.4.10) Jx (X r S =-#H (S {U (X+ # S ) +U (x+#s) (U (x)-U (x+#s) ) )

+ ; L O (xrs)

with E C ( l R + C o ) n C ( I R + L2) and

? ' ' O ( x , s ) = o for s < o

Hence N is differentiable with respect to x and

Nx E C o ( e ) .

Let us now consider differentiation of J with respect

to the second variable s. Since differentiation D

is a continuous linear operator from 8' into G'

and Vv E GI: DFv = F(-ikv) (Yosida, 1974, pg. 151-152)

it is easy to see that Js is well-defined and:

(4.4.11)

In order to obtain more information about Js we

observe, that:

Js = (2n)-#F2(-ik(L-l)) E C ( I R + G I )

m

-ik(L-1) = -ikH1 - ikH2 - ik 2 Hn. n=3

Easy calculations show, that:

a F2(-ikH1) = =(F2Hl), i.e.

(F2(-ikH1) (x,s) = 4 ${U(x)6(s) - u(x+fs)H(s)) 6 ( s ) denotes Dirac's delta functional.

-ikHZ E C ( I R + L2), i.e. F2(-ikH2) E C ( l R + L2)

F2(-ikH2) = # J 3 ( s ) [U(x+#s)lU(x)-U(x+#s) 1

-

-

+ r" u(y)U(y+#s)dyI m X

I i

W

E C ( I R +L1 nL2) ,i.e. F2(-ik 2 H ) E C ( I R +Co nL2) n=3 n=3 n

(x,s) = 0 Vx E IR and Vs E x-.


The conclusion is that J -#U(X)~(S) E C(B -+ L ~ ) and

(4.4.12) [Js-#U(x)6(s)l(x,s) = ? o t l ( ~ , ~ ) + S

As a consequeLce N is differentiable with respect to s

and Ns E Co (IR+) . 2

(iii)

we shall now derive an integro-differential equation

f o r N. For that purpose we rewrite @.2.1.l).(ii) as

f 01 lows : m

-(L-l)'(x,-) = 2ik(L-1) (x,-)+lu(y) (L-1) (y,')dy+U(x). X

m

The integral I u(y) (L-1) (y,.)dy converges in L2-sense. When we apply (27r)"F to both sides of this equation,

we find:

X

m

-J (x,.) = -2JS(x,.) + u(y)J(y,.)dy+U(x)6(s). X

X

This leads us to the following problem for N(x,s),

s 2 0:

N(x,O) = #u(X)

lim SU~~N(X,S)~ = o X+= S X

In fact (4.4.13)constitutes an hyperbolic boundary value

problem of Goursat type for N.


The boundary condition at s = 0 is imposed, because

of 4.4.8. The condition for x --* is also rather

obvious. This can be seen as follows : 2 Using the inequality I Z Hn(x,k) lG 2 minc[I lu(y) Iydyl ,

[+I lu(y) Idyl 1 valid for x sufficiently large > 0 (see (4.2.2.8)and (4.2.5.2) we see that

in L1 for x + m.

Hence F c Hn(x,-) + 0 in Co f o r x + m .

n>2 In combination with(4.4.9) this yields the condition

for x + m.

m

n>2 X UlkI 2

X Z Hn(x,-) --* 0 n22

In order to investigate the properties of N we reformulate (4.4.13) as an integral equation:

(4.4.14) N(x,s) = #U(x+#s)+#l I (uN) (y,n)dydn

This integral equation can be solved iteratively

(4.4.15)

s m

0 x+#s-#n

m

N = E N n n= 0

NO(x,s) = fU(x+fs),

With induction with respect to n one can show that

(4.4.16) (N,(X,~) I G f. Vo (x+fs), n 2 0 (V, (XI +x-vo (XI 1

n! m m

with Vo(z) = I lu(y) ldy, Vl(z) = I lU(Y) I lyldy. These estimates are not completely trivial. Let us demonstrate

Z

4.4.16) is satisfied for the induction step. Of course n = 0.

If x E lR and s > 0, then with M(y) = Vl(y) + x-. Vo(y):

the notation


We conclude that this iteratively constructed solution

of (4.4.14)is an element of C(R+), which satisfies the

estimate :

2

It is left to the reader to show that the homogeneous

equation corresponding to (4.4.14) m S

v(x,s) = 4 I I (uv)(y,o)dydu 0 x+#s-#ll -

2 possesses only the trivial solution in W = { v E C ( l R + ) ] tla E R sup sup Iv(x,s)I < m}

As a consequence the integral equation (4.4.14)is

uniquely solvable in W and N as constructed in (4.4.15)

coincides with what we called N previously.

Now (4.4.17) implies that I 0

function of x E 1R. Hence J as defined in (4.4.6) is

indeed E C(R + L1) and the proof of theorem 4.4. is complete.

x>a s>O

m

IN(x,s) Ids is a continuous

0


The transcription of theorem 4.4 in terms of Jig is given below.

Corollary to theorem 4.4. The s o l u t i o n J I , o f t h e

S c h r d d i n g e r e q u a t i o n h a s t h e f o l l o w i n g F o u r i e r

r e p r e s e n t a t i o n :

where t h e kerneZ K i s r e l a t e d t o N g i v e n i n 4.4.7 by:

The p o t e n t i a l u i s f o u n d from K by t h e r u l e :

(4.4.19) d dx U(X) = -2-K(x,x).

Of course (4.4.18-19) are immediate consequences of

JI,(x,k) = L(x,k)eikx and theorem 4.4.

Let us conclude this section with the following result

which will prove its usefulness further on.

Lemma 4.2. The e x p r e s s i o n s g i v e n in(4.4.7)and(4.4.18)

a r e v a l i d for a l l (x,k) E R x q, i f t h e p o t e n t i a l u

s a t i s f i e s a g r o w t h c o n d i t i o n of o r d e r 1.

Proof of lemma 4.2. Define: v (x, k) =L (x, k) -1-JN (x , s ) eiksds.

For each x E IR the function v(x,k) is analytic in k on

C, and continuous in k on q. Further (4.4.7) implies:

OD

0

(4.4.20) v(x,k) = 0 V k E R.

We shall also show, that:

(4.4.21)

It is not difficult to demonstrate that consequently:

71

lim I Iv(x,reie) Id0 = 0. r+- 0

(4.4.22) v(x,k) = 0 Vk E C,.


Namely, using Cauchy's formula we find for k E C+:

Of course the contents of (4.4.20-22)are that (4.4.6)

holds for all (x,k) E IR x C+. -

In order to prove(4.4.21)we proceed as follows.

Using the asymptotics of L given in theorem 4.2.4

it is clear, that lim [ IL-11 (x,reie)d8 = 0 r-

As for I its absolute value by

I T m N(x,s)e irscos8-rssinedsd8 we can estimate

0 0

1 Now we choose E(r) such that sin 4 2 - for all 4 [E(~),IT-E(~)] and sin(e(r)) = -. Note that E (r) S= - for r -, a.

In this way we obtain the estimate

1 J 3

J3 J3

I T m Hence: lim I 11 N(x,s)eirscose-rssinedslde = 0.

By the definition of v it will now be clear that

(4.4.21) is valid.

r- 0 0

It is left to the reader to do the simple verification

of (4.4.18)for k E C,. 0

4.5. INVERSE SCATTERING

In this section we shall show, that the kernel N(x,s)

(or K(x,s)) introduced in (4.4.7),(4.4.18) has to satisfy


a uniquely solvable integral equation, in which the

coefficients are completely determined by the scattering

data. As a consequence of the relation between the

kernel N (or K) and the potential u given in(4.4.8) (or

(4.4.19)) this integral equation for N (or K) provides

us with a procedure to determine the potential of the

SchrGdinger equation from the scattering data. This

procedure is refered to in the literature as inverse

scattering., transformation (or transform) . The integral equation mentioned above bears the names of

Gel'fand, Levitan and Marchenko. In Gel'fand, Levitan,

1951 the essential ideas of inverse scattering were

already developed.

However these authors considered the SchrBdinger equation

on a half line and formulated an integral equation with

domain of integration containing the finite endpoint of

the half-line. Later Marchenko, 1955 and Faddeev, 1959

considered the inverse scattering on all or W.

It is their form of the integral equation, which we shall

present here.

In this section we shall assume, that the potential u

satisfies a growth condition of order 2 . In theorem

4.3.11 we saw that then the number of discrete eigenvalues

is finite, say d m . We shall denote these eigenvalues

by - p l < ... < -pd < 0 and we introduce kn = x, 1 G n G d.

Let us also introduce:

discr + Bcont' B = B

The coefficients Cn are defined as follows. Let JI,, 1 < n =G d be the real eigenfunction, normalized to 1 in

L -sense and positive for x + - O D , corresponding to the

discrete eigenvalue -pn, 1 G n G d. Then: 2


knx (4.5.2) Cn = lim [$(x)e

Note that the definition of Cn given here is in complete

accordance with the one given in chapter 2.

It is a simple exercise to show, that Cn can be

identified with a(i$)/ll+,(-,i$)ll with a(k) as in

corollary 4.3.1, i.e. the

by the relation qr(*,i%) = Cn+& ( ~ , i ~ ) ~ l l + r ( - , i ~ ) ! l .

X'm

constant Cn can also be defined

If the discrete spectrum is void, i.e. d = 0, then we

define of course Bdiscr = 0.

In the definition of Bcont we denote by br the reflection

coefficient introduced in (4.1.18) . Let us demonstrate that br is an element of L2. It is an immediate consequence

of the identity lar[ + lb,l = 1 that Ibr(k) I Q 1 on IR

(see 4.1.23). Using the asymptotics for Ikl + m of ar(k)

= r-(k)-' = 1+(2ik)-l.? u(y)dy + O(lkl-2)

2

(see 4.2.4.5) - m

we find, that Ibr(k) I = l-lar(k) l 2 = O(1kl-l) for k + 03.

The conclusion is that indeed br E L2 and consequently

also Bcont E L2 and B E L2.

We can express B as follows

m 1 d 2 - k ~

(4.5.2) B(z) = Cne n + 5 J br(k)eikzdk, z 2 0. n= 1 -03

W 1 Here one has to interprete J br(k)eikzdk as

(2n)-#(F-'br) ( 2 ) . -a

Note that B(z) depends only on the scatterinq

data br ,knICn, 1 Q n

The main result is now given by:

d and that B is real (see 4.1.21).

Theorem 4.5.1. If t h e p o t e n t i a Z u s a t i s f i e s a 2nd o r d e r

growth c o n d i t i o n , t h e n :

(i I

, m ) ) i n t r o d u c e d i n t h e k e r n e l N E Co(IR+ ) fl C(IR + L1(o -

2


(4.4.6) s a t i s f i e s t h e f o l l o w i n g r e a l i n t e g r a l e q u a t i o n

f o r x E IR , s 2 0: m

(4.5.3)

l i i ) t h e s o l u t i o n of 4.5.3 i s u n i q u e i n F(IR + L2(0,m)).

Herewith the inverse scattering method is well-founded!

Given the scattering data br; Cn,kdlG n G d one constructs

the function defined in (4.5.22. Next solving the

integral equation (4.5.3) determines N uniquely.

The potential u is then founds asu(x) = -2Nx(x,0),

see (4.4.8).

0 = B (2x+s ) +N (x, s ) +IN (x, t) B (2x+s+t) dt 0

Proof of theorem 4.5.1.

(i)

The proof of (i) is based on the simple identity (4.1.19)

arqr = brqR + 5,. -ikx ikx Substitution of the formula's qr = Re

(see (4.1.2-6)) leads us to , $& = Le

2ikx + x. arR = brLe

Next we use the Fourier expression giveh in (4.4.5):

L = 1 + GF;lJ, where J is the function given by

(4.4.6): J(x,s). = 0 for s < 0, J(x,s) = N(x,s) for s 2 0.

In this way we find: -

arR-1 = bre 2ikx +. fibre2ikxp;1J 4- mFilJ. -1 Since J is real we have: F2 J = F2J. Hence:

-(arR-1) 1 = -b 1 e2ikx + bre2ikxFi1J + F2J. fi G r

It is easy to verify, that each term in this identity is

an element'of C ( I R + L~).

As a consequence we can have F 2 to obtain the following identity:

-1 operate to both sides

134 W. ECKHAUS 5 A. VAN HARTEN

-F2 1 -1 (arR-l) = -F2 1 J2n J2a

-1 (b e 2ikx) r (4.5.4)

+ F;’ (bre2ikxF;1J)

+ J

Each term in 4.5.4 is again an element of C ( 1 R -+ L2)

(see 4.4.4).

We shall now show, that for s 2 0

(2x+s (4.5.5) 9: - { ( 2 ~ ) F 2 (bre2ikx) 1 (x,s) = Bcont - 4 -1

m

b: {F-l (bre2ikxF,1J) 1 (x,s)=.fN(x, t)Bcont (2x+s+t)dt - 2 0

Of course the integral equation(4.5.3)is an immediate

consequence of (4.5.4-5).

It is left as an excercise to the reader to show, that

B is real (br(k) = br(-k) , see (4.1.20)).

In fact(4.5.5)~ - and B can rather easily be derived, but the derivation of(4.5.5)c - is somewhat more difficult.

ad 4.5.5-a

= (27~) -4 F -1 br E E L2, L1 fl L2, br = lim E J . 0 b: in L2, BEont

= lim Bkont. We denote further by Tal a E IR € 4 0 Bcont

the translation operator defined by:

A simple calculation shows, that:

(Taf) (x) = f (x-a) . m

{ ( 2 ~ ) - # F2 -1 (bre E 2ikx ) 1 (xls)=(2a)-l .f b:(k)eik(s+2x)dk =

-m

EQUATION 135

in L2-sense and

(2n)-’Fi1 (b;eSikx) 1 (x, - ) in L2-sense we obtain by taking the limit E S O :

{ (2rr)-’Fi1 (bre2ikx) 1 (x, 1 = T-2xBcont.

Herewith the contents of p have been demonstrated. -

ad 4.5.5-b.

A simple calculation shows, that: -__w1s

m m { F i l (b:e2ikXF;1J) } (x,s)=(2a)-’ J bE(k)e2ikx.JN(x,t)eiktdt.eikb -m 0

m

Here we have used, when interchanging the order of

integration, the fact that all integrals converge

absolutely.

Taking the limit EJ.O and usingthe fact that both sides

converge for fixed x E IR in L2-sense with respect

to the variable s, we find (4.5.5) b . -

ad 4.5.5-c.

1 -1 Define I(x,s) ,={-F2 (arR-l)}(x,s) and ---_MI

LE m

I (x,s) = & J (arR-1) (x,k) (l-iek)-’eiksdk, E > 0. -m E

Then, if we restrict ourselves to positive values of s,

we have

I ( x , . ) = lim IE(x,.) in L2(0,m)-sense. E + O

Note that the integral defining IE converges absolutely.

We further bbserve, that the integrand (arR-l) (x,k) - (1-iEkI-l -eiks has a number of nice properties.

This integrand is a meromorphic function of k on C,


with a finite number of poles of order 1 in the points

i$,.. . ,i% (see theorem 4.2.3 and theorem 4.3.1

(ar = r- 1 ) . Moreover the integrand is continuous in k on c,, because of the 2nd order growth condition and theorem 4.2.5.1 and its corollary. If s 2 0 then it decays as O(lk1'2) for

- 1

Ikl-+-,k E F+. As a consequence we have :

where hM is the contour consisting of [-MIMI c IR and half a circle around the origin with radius M.

An application of Cauchy's theorem of residuals

shows I that:

Taking the limit E + O we obtain:

(4.5.6) I(x,s) = i C dr e

Using (4.3.18)and the relation Cn = a (ix) /I1 $, ( I i x ) II (see the exercise just below(4.5.2))we can derive

that dr- dk ( i s ) = ( i c n ) - l ~ ~ + r ( - I i ~ ) ~ ~ . Next we express R(x,ix) in L(x,ix). This is done

by applying the relation Cn+e (x,i%) =

$r (x, i $ ) / l l $r ( * I ix) I1 (see again the exercise just

below (4.5.2).

We find: R(x,i<) = C,II$r('li~)IIL(x,i~)e

Hence:

d R(x,i$) -kns

n=l -- (iJlln) dk

-2knx

Finally we substitute the identity L(x,iG) = m -knt n

1 + I N(x,t)e dt (see 4.4.7 and lemma 4.2) 0

into (4.5.7). This yields :


d -kn(s+2x) I(X,S) = - I: Cne

n= 1

1 dt 2 -kn(s+t+2x) m

- I N(x,t) ( I: Cne 0 n= 1

The conclusion is that(4.5.5-c)indeed holds.

(ii)

We start the proof of (ii) with the observation, that

the integral equation (4.5.3) depends parametrically on

the variable x.

Let us define

following way:

(4.5.8) (B(x)f) ( s ) = f (t)B(s+t+Zx)dt 0

Suppose that there exist two different solutions of

4.5.3 in F(7R + L2(0,m)). Then this would imply that

for some x E IR the equation

(4.5.9)

as an operator on L2 (0,~) in the

m

E W f + f = 0

has a non-trivial real solution in L2(0,m).

However we shall show, that this is impossible.

In order to do this we shall first introduce some

notation. L2(0,m) is the space of

functions on (0,~).

For f E L2 (0 ,031 we denote by f the extension of f to

(-m,m),which equals 0 on (-m,O), i.e. f(y) = f(y) on

( 0 , ~ ) and ?(y) = 0 on (-m,O).

The notation of the innerproduct on L2(0,-) and the

corresponding norm is ( I ) , 11 11. By definition e; E Lz (0 ,m) is the element for which

en (y) = Cnexp (-kn (x+y) 1.

square integrable

- N

X

The following equality will play a fundamental role in

the proof of the unique solvability of (4.5.9).


m

+ I (F-lG) (k). (F-’?) (k) .br(k)e2ikxdk -m

If g and f are elements of L2(0,m) n Ll(O,m) then the proof of 4.5.10 is easy:

(g,B(X)f) =

d m -kn ( S+X) m -kn (t+x) = z C: . I g(s)e ds I f(t1.e dt

+ I (2a)-’. I G(s)eiksds. (2a)-’.f ?(t)eiktdt.br(k

n= 1 0 0 m m m

-m -m -m

= right-hand side of(4.5.10).

In this calculation all integrals converge abso

since Ibr(k)I < 1, Vk E IR (see(4.1.22)).

e2ikxdk =

utely,

Using that L2(0,m) n L1(O,m) is dense in L2(0,m) it is clear that(4.5.lO)holds for all g E L2(0,m) and

Hence for f E L2(0,m) n Ll(O,m) we have demonstrated, that B f is a continuous linear functional on L, (0, m)

f E L2(0,m) n L1 (0,m).

L d

n= 1 with a norm < II fll ( Z I1 eEll + 1). This implies that B(x)f can be identified with an element of L2(O,m), which

satisfies the estimate tIB(X)fII < (1+ Z t ~ e ~ t ~ 2 ) - ~ ~ f ~ ~ . The

conclusion is that is a continuous operator from

L2(0,m) into itself and that (4.5.10) holds for all

d

n= 1

f,g E L*(O,m).

Now suppose that f is a non-trivial solution of (4.5.9).

Using (4.5.10) we find

(4.5.11) 0 = (f,B(X)f + f) = d m

n= 1 -m = (f,f) + Z (f,e:)*+j E(F-’?) (k)}2br(k)e2ikxdk


m m

Since (f,f) = J [:(XI I2dx = I I (F'l:) (k) I2dk

we obtain the estimate -m -m

d m

(4.5.12) 0 C l(f,e;)l2 + I I (F-lF) (k) 12(1-

Because of the fact that r-(k) cannot have zero's

l R \ { O j and . given (4.1.22) we have,that l-lbr(k)

on l R \ { O l . The conclusion is, that

n= 1 -m

This immediately implies, that 2 = 0 E L2 and also

that f = 0 E L2(0,m). Hence(4.5.9)cannot have non-

trivial real solutions in L2(0,m). 0

Let us conclude this section with the transcription

of Theorem 4.5.1 in terms of the kernel K.

Theorem 4.5.11. If t h e p o t e n t i a l u s a t i s f i e s a 2nd o r d e r

growth c o n d i t i o n t h e n t h e k e r n e l K i n t r o d u c e d in(4.4.18)

which i s r e l a t e d t o N by K(x,s) = N(x,s-x) i s t h e u n i q u e

s o l u t i o n w i t h t h e p r o p e r t y K(x,-) E L2(xIm) , Vx E IR

o f t h e i n t e g r a l e q u a t i o n

(4.5.13) 0 = B(x, s)+K (x,s)+.fK (x, t)B (t+s)dt.

T h i s j u s t i f i e s t h e i n v e r s e s c a t t e r i n g method a s

e x p l a i n e d i n c h a p t e r 2 .

m

X

4.6. CONCLUDING REMARKS.

It should be mentioned that the method of direct and inverse

scattering transformation can be generalized to the so called matrix

SchrGdinger equation, see Calogero and Degasperis, 1978. There

also exists an extension of the scattering theory to the

SchrGdinger equation with a potential which grows linearly in

x for l x I + m, see Calogero, Degasperis, 1978. A generalization

in a different direction will be considered in the next

chapter. Furtherimre, it is sametimes possible to derive sharper


results than given in this chapter under more restrictive

assumption. For example : exponentially decaying potentials

or potentials with a compact support lead to scattering data

which are analytic in larger domains, see Ablowitz, etal,

1974 and potentials which are analytic in a strip around the

real axis give rise to exponentially decaying reflection

coefficients, see Trubowitz, Deift, 1979. Finally we remark

that there also exists a scattering theory for the SchrGdinger

equation with a periodic potential, see Trubowitz, 1977 and

Dubrovh, Matvew, Nobikov, 1976. This theory is quite different

from the one presented here.

CHAPTER 5

DIRECT AND INVERSE SCATTERING FOR THE

GENERALIZED ZAKHAROV-SHABAT SYSTEM

A theory for direct and inverse scattering analogous to the one

developed in the preceeding chapter has been developed for

several other systems of ordinary differential equations. We

shall describe here how this can be done in the case of the

generalized Zakharov-Shabat system. This system gives rise to a

rich scala of integrable non-linear evolution equations, see

chapter 6. It has the following form:

V I -ir; q v1

2 v2 (5-1) = (r ) In this system there are two complex potentials q and r and the

constant 5 is the spectral parameter. A restricted form of

(5.1) was studied by Zakharov-Shabat, 1972. Later on a genera-

lized form of the original system got a lot of attention, see

Ablowitz, et al., 1978, 1979. However, the scattering theory

for the generalized Zakharov-Shabat system is still not as

complete as the scattering theory for the SchrBdinger equation.

This will become clear in our analysis.

It is easy to verify, that the Schradinger

(4.1) can be identified with the following

(5.1): v1 = J I , v2 = JI ' + ik+ with X = k , 2

equation given in

special case of

5 = k, q 1, r = u.

However, the conditions under which we shall study the

scattering problem for (5.1) are such that the case q I corresponding to the Schradinger equation is excluded.

141


Of course the conditions that are needed for the analysis of

the scattering problem for (5.1) are growth conditions on the

potentials q and r. In what follows it will be sufficient to

suppose that

(5.2.) q,r E C 1 ( I R )

and moreover:

(5.4) llqll < m , IIrII < - L. L.

In contrast with (4.2-3-4) we have now also requirements on

the derivatives of the potentials. These requirements play a

role in the proof of the asymptotics of the solutions for 5 -+

(see theorem 5.2.6) .

It is somewhat surprising, that we do not need 2nd order growth conditions on the potentials as in the case of the SchrBdinger

equation, see (4.4). We remind the reader, that in the case of

the SchrGdinqer equation (corresponding to q 1 in (5.1)) the

stronger growth condition on the potential u (i.e. r in (5.1))

was necessary in order to prove continuity of the transmission

coefficient at k = 0 (i.e. 5 = 0). It will appear, that the

requirement Jlq(x) Idx < m in (5.4) has the effect that no

special singularities at 5 = 0 will be mesent.

In this respect the analysis in this chapter is considerably

easier than in chapter 4. However, there is also a complication

which is present here, but not in the case of the Schradinger

equation: the differential operator, which appears in the

generalized Zakharov-Shabat system is generally not self-adjoint.

m

-m

The organization of this chapter is the same as in chapter 4.

In section 5.1 we shall introduce some families of solutions

ZAKHAROV-SHABAT SYSTEM 1 4 3

of ( 5 . 1 ) and define some scatterinq coefficients. Next in

section 5 . 2 we consider the regularity and asymatotic behaviour

of the solutions of ( 5 . 1 ) . In section 5 . 3 we consider the

spectrum of the generalized Zakharov-Shabat operator and in

section 5 . 4 we give Fourier integral expressions for some

solutions of ( 5 . 1 ) . In section 5 . 5 we shall derive the analogue

of the Gel'fand-Levitan-Plarchenko integral equation associated

with the inverse scatterinq problem for the generalized

Zakharov-Shabat system.

It can be shown, that this integral equation is uniquely solva-

ble if r = -q*, see Ablowitz et al, 1 9 7 4 . However, in the

general situation the problem of sufficient conditions for

unique solvability covering a wide class of Dotentials is still

open (Ablowitz, 1 9 7 8 , p. 3 4 ) . Note, that this is quite diffe-

rent from what we found in the case of G.L.M. inteqral equation

corresponding to the SchrGdinger equation (see theorem 4.5.1).

It is probably true that this difference is caused, at least

partly, by the non-selfadjointness of the generalized Zakharov-

Shabat operator.

In what follows various proofs are omitted or only outlined

because they run parallel and quite analogous to the proof of

the corresponding theorems in chapter 4 . Filling in the details

is therefore left as an exercise to the reader. Furthermore,

we remark that we use in this chapter a notation for the

scattering dat.a, which is analogous to what we used in chapter

4 . This notation is different from the one used in chapter 7

and in Ablowitz, 1 9 7 8 . Further on we shall specify the

relation between these different notations, when we introduce

the scattering coefficients.

5 . 1 . SOLUTIONS AND SCATTERING COEFFICIENTS OF THE GENERALIZED

ZAKHAROV-SHABAT SYSTEM.

It is quite'natural to consider the following 4 families of

solutions of the generalized Zakharov-Shabat system ( 5 . 1 ) .


-i<x (5.1.1) (a) $r = Re

Here $,, R, etc. denote vector functions with two components.

The relation between the notation introduced in (5.1.1) and the

one of Ablowitz, 1978 and chapter 6, 7 is as follows: $ = - - z - 1:

$r = = $ , 9 , = $. Let us further introduce the notation:

(5.1.2) Q(x) = (O q(x)) , r(x) 0

-2i 0

- Then R, z, L and L have to satisfy:

(5.1.3) (a) R' = (<M+ + 9)R , lim R(x,<) =

(b) R' = (gM- + Q ) i i , lim R(x,<) =

(c) L' = (CM- + Q)L , lim L(x,?) =

X+-m - u

X+-m

X*m

In section 5.2 we shall show, that the problems in (5.1.3) are

uniquely solvable, if the location of the spectral parameter

is suitably restricted.

(5.1.4) RIL and $r,$,

R,L and qrITk are well-defined f o r Im 5 3 0

are well-defined f o r Irn 5 < 0 - -

ZAKHAROV-SHABAT SYSTEM 145

Note that for 5 EIR we have defined f! solutions of the 2nd

order system (5.1). It is clear, that $r and $r are linearly

independent and so are $& and $&. Hence for real 5 we musthave

- -

5 N

(5.1.5) a. qr = r+$R+r-$R I b. q R = R-$,+R+$, - 5 5 - - 5 - - qr = r++R+r-$R JI, = R-JIr+R+$,

with scattering coefficients r+, r-, etc. only dependent of

5 EIR. The notation o f the scattering coefficients introduced

in (5.1.5.a) has the following relation to the ones used in

Ablowitz, 1978 and chapter 6, 7: a = r-, b = r+, a = r

b = -r-. Of course there are many relations between r+, r-,

r+, r-, R - , R + , R-, R+. Some of them are specified here below:

- 5

+' - - - 5 - -

5 - ?r r - r-r+ = 1 - + (5.1.6) (a)

(7 r+

r-,-l R 1 -

j, i.e. - - &+ R- = -r-, 11, = -r+

The relation (5.1.6-a) can be demonstrated by considering the

and checking that W($=,Gr) is constant. The relation (5.1.6-b)

is elementary.

Consideration of other Wronskians leads to some more very use-

ful results. It is easy to verify that W($r,@R) = W(R,L) is

well-defined for 5 E , independent of x and for 5 El?? equal to r-. Analogously W($r,$k) = W(g,E) is well-defined for

5 E z-, indepedent of x and for 5 EIR equal to -r+.

Hence, it is natural to extend the domain of r_ from IR to e+ and to extend the domain of r+ from IR to z- by the following

definitions.

-+ 5

N

- (5.1.7) r- = R1L2 - R2L1 on C+

r+ = LIRl - L2R1 on E- def

de f

- 5 - 5 -

Note, that these definitions do not give rise to a potential

singularity at 5 = 0 (compare (4.1.16)!)


Finally we remark that the following property holds when q and

r are real: replacing 5 by -5 in +,,9,,$,,+,, R,R,L,L,r+,r-,

r+,r- is equivalent to taking the complex conjugate. Hence,

for example we then have:

- - - - - -

Consequently all these functions are then real if Re 5 = 0. The

verification of these statements is trivial.

5.2. PROPERTIES OF SOLUTIONS

To start with we shall reformulate the problems for R, R, L,

given in (5.1.3) as inteqral equations. Next we shall solve

these inteqral equations iteratively and consider the regulari-

ty and the asymptotic behaviour of the solutions.

In order to formulate the integral equations in a conveniently

abbreviated form we shall aqain use the notation * and * introduced in (4.2.2.2), but now extended in the obvious way

to matrix kernels S(x,y,c) and vector functions w(x,<). It is

easy to check that the problems given in (5.1.3) are equivalent

to

-

- V

(5.2.1) (a) = + G ; R (c) L=(:)-HiL

For example (5.2.1-a) is an alternative notation for the

following two coupled integral equations


Analogous to theorem 4.2.2 we can now derive the followinq

results :

Theorem 5.2.1. Suppose t h a t t h e p o t e n t i a l s q,r s a t i s f y

(5.2,3,4). Then:

(il a . t h e r e i s a u n i q u e s o l u t i o n R of ( 5 . 2 . 1 - a ) f o r 5 E z+, which i s c o n t i n u o u s in x and bounded f o r x + -m.

R s a t i s f i e s ( 5 . 2 . 3 - a ) i n c l a s s i c a l s e n s e .

b . t h e r e is a u n i q u e s o l u t i o n of (5.2.1-b) f o r 5 E z-, which i s c o n t i n u o u s i n x and bounded for x + -m.

R s a t i s f i e s (5.2.3-b) i n c Z a s s i c a 1 s e n s e . -

c . t h e r e i s a u n i q u e s o l u t i o n L of ( 5 . 2 . 1 - c ) f o r 5 E z+, which is c o n t i n u o u s i n x and bounded f o r x -+ +a.

L s a t i s f i e s ( 5 . 2 , 3-c) i n c l a s s i c a t s e n s e .

d . t h e r e i s a u n i q u e s o l u t i o n of (5.2.2-d) f o r 5 E E - , which i s c o n t i n u o u s i n x and bounded f o r x -+ +m.

L s a t i s f i e s ( 5 . 1 . 3 - d ) i n c l a s s i c a l s e n s e . -

( i i ) t h e s e s o l u t i o n s can be g i v e n a s c o n v e r g i n g Neurnan s e r i e s :

m

(5.2.2) ( a ) I? = Rn n=O

(c) L =

L(O

m

z L(n) n= 0

148 W. ECKHAUS I% A. VAN HARTEN

The speed of c o n v e r g e n c e is of f a c t o r i a Z t y p e a c c o r d i n g

t o t h e foZZowing e s t i m a t e s

where f o r e E C2 : /el = max(lell,le21) and

Comparing the estimates in ( 5 . 2 . 3 ) with those of (4.2.2.6-8)

there is one most significant difference: the singularity at

k = 0 ( 5 = 0)is no longer present: This fact allows us to

give a regularity result analogous to theorem 4 . 2 . 3 , which has

the advantage that 5 = 0 is no longer an exceptional case.

Theorem 5 . 2 . 3 .

(i) R and L a r e eZemen t s of C( IRxC+) and a r e b o t h anaZytic i n

5 on C f o r each x Em. and L a r e eZemen t s o f C(lRxz-) and a r e b o t h a n a Z y t i c i n

5 on C- f o r each x Em.

;t liil


As an immediate consequence of (5.1.7) we now also have the

following:

- Corollary to theorem 5.2.3.

li) r- is c o n t i n u o u s on C+ and a n a Z y t i c o n C,.

(ii) r+ is c o n t i n u o u s on C- and a n a l y t i c on C-.

- *

Another consequence of the estimates given in (5.2.3) is that

R,L and z r E are uniformly bounded on their natural domains IR x z + r IR x E - .

Comparing with theorem 4.2.5.11 we notice that in this

situation no linear growth of IRI I ILI I / E l , IzI in x for x +

or x -+ --m can arise.

We shall now consider the asymptotics of the solutions for

1x1 -+ in more detail.


Part (i) of this theorem is again an immediate consequence of

the estimate in (5.2.3). For example for x negative and 1x1

sufficiently large it follows, that lR(x,~)-(~) I 2U0(x) and

this implies (i)-a. The contents of part (ii) are found by

combining (i), (5.2.4) and (5.1.7). Next, part (iii) is found

by using a technique as in the proof of theorem 4 . 2 . 4 , part b.

Finally, part (iv) follows by using (5.1.5) and (i) .

1

Using (5.2.1) it is easy to see that the following result

holds :


This corollary is quite useful. For example it implies at once

that r+, r- E C( IR) and u

with B as in (5.2.4).

The reader will have noticed, that by now we have found ana-

logues of theorem 4.2.4, part b and the lemma's 4.2.4.1-2, but

no analogue of theorem 4.2.4, part a, where the asymptotics

of the solutions for Ikl + m was given. The reason is that in

the case of the Schradinger equation the nth term in the

Neuman series of the solution is of the order Ikl-", see

theorem 4.2.2, whereas this is no longer automatically the

case here, see (5.2.3). Though this fact was very pleasant

before (no singularity at 5 = O ) , it works now in the wrong

direction: we have to work harder to'get the asymptotics for

I r ; l + m. Let us consider the asymptotics of R ( x , 5 ) for 1 ' 1 + m,

5 E ?+. First we shall use a formal expansion procedure to find the first few asymptotical terms. Next we shall prove the

correctness of the formal expansion.

We look for an expansion of the following form

0 R(x,s) = Ra

Subsitution in (5.2.1-a)

x ) + 1 Ra 1 ( X I < ) + 00-2. 1

5 5

leads immediately to:

Rao(x) = (i)


For the next order we obtain

-m

It is easy to check by integration by parts, that:

X

dY) J r(y)e 2i<(x-Y)dy = -- 2ic{r(z)- 1 I r' (y)e 2ic (x-Y) X

- W -m

By substitution of this identity and applying integration

by parts we find

l X 2iz; -m

2i<(y-z)dzdy = -- I q(y)r(y)dy + I q(y) I r(z)e

q' (y) Ir' (z)e2ic(yz'dzdy

Y X

-m -m

1 X 2 i r , (x-z) X Y dz - + -T[q(x) I r'(z)e

- W -02 -m X

(2i~)

- I q(y)r' (yldyl. -m

Hence it seems logical to take

Note that the conditions on the potential r given in (5.2,3,4)

do not imply that I r' (y)e

Of course this would be the case if we would integrate by

parts another time, i.e. if r E C (m) and I1r"II < m .

1 5

X 2i<(x-y)dy = 0 ( 1 - 1 ) .

-m

2

L1

Let us now consider the remainder term

R r = R - R a - - Ra' z;

It is not difficult to check that Rr has to satisfy the

following integral equation - Rr = h + G*Rr

with


-m -m -m

N o t e t h a t :

w i th a c o n s t a n t C > 0 independent of x EIR. By s o l v i n g t h e i n t e g r a l e q u a t i o n €or t h e remainder t e r m i tera- t i v e l y w e see, t h a t t h i s f a c t i m p l i e s t h a t

w i th a c o n s t a n t C1 > 0 independent o f x EIR!

Th i s proves t h e c o r r e c t n e s s o f t h e expans ion f o r R. I n a n analogous way w e can f i n d t h e a sympto t i c s o f t h e o t h e r f a m i l i e s of s o l u t i o n s f o r 5 + m. The r e s u l t s a r e g i v e n i n t h e fo l lowing theorem.

Theorem 5 .2 .6 . Suppose t h a t t h e c o n d i t i o n s on t h e p o t e n t i a l s

g i v e n i n ( 5 . 2 , 3 , 4 ) a r e s a t i s f i e d . The a s y m p t o t i c f o r Zarge

i s t h e n g i v e n by X

Is ( y ) r (y ) dy

dY 2 i C (x-y) (5 .2 .10) ( a ) R ( x , < ) = (i) - & (-m

r ( X I - J r ' ( y ) e -m


T h e o r d e r symboZ 0 is v a l i d u n i f o r m Z y in x E IR.

Using theorem 5.2.5-ii and (5 .2 .8 ) we can also derive expres-

sions for the behaviour of the scattering coefficients for large 5.

Corollary to theorem 5.2.6.

f o r I 5 1 -b m , r; E z+


and IIr-II < L1 -

The derivation of these results for r+ and r- are completely

trivial. For example it follows from ( 5 . 2 . 8 ) by integration by

parts, that

Using the theory of Fourier transform and ( 5 . 2 , 3 , 4 ) we see that

s ( s ) =

with the property lim I s ( < ) [ = 0. It is also clear, that

- s ( 5 ) is absolutely integrable on each interval (a,-) or c (-,-a) with a > 0, for both - and s are in L 2 ( a , m ) and L2 L2 (--,-a).

a,

I r'(y)e- 2iz;ydy defines and element of C( m) L2 ( IR) -m

1 111'"

1 c

2 i O d 5.3. THE SPECTRUM OF ( o -i)(x - Q) ON L 2 ( I R )

Let us consider the operator L, with domain

The Zakharov-Shabat system (5 .1 ) can also be formulated as:

( 5 . 3 . 1 ) Lv = 5v

and is therefore intimately related to the spectrum of the

operator L. Note that L is an unbounded, closed operator on L, ( IRI2. Moreover L i s a bounded perturbation of the self-

L

adjoint operator since the potentials q and r are

bounded on lR. However, in general L is not selfadjoint.'

As before we define the resolvent set p(L) by 2

p (L) = { r ; E d l L - 5 has a bounded inverse on L2( IR) 1. IPa(L)

is the set of all isolated eigenvalues. Furthermore,

(J ( L ) = 5 E C I L - 5 is not boundedly invertible on L 2 ( IR) 1 . 2


We now find the following characterization of p ( L ) o(L) and

IPa(L) in terms of the scattering data:

The proof is analogous to the proof of theorem 4.3.1. We also

obtain a corollary analogous corollary 4.3.1.

Corollary 5.3.1. Each e i g e n v a l u e E IPa(L) i s s i m p l e . The one -

d i m e n s i o n a l e i g e n s p a c e c o r r e s p o n d i n g t o 5 € IPo(L) is spanned *

+ and by if 5 E C . Fur thermore - by UJr(-,r) if 5 : Q: - $ ( s t < ) = a(C)$k(*,C), if E IPa(L) n C+ and $ r ( * r ~ ) = ,r

a r e r e a l t h e n < E I~O(L) =. - 5 E IPU(L) . if 5 E IPa(L) n C - . Moreover, if t h e p o t e n t i a t s -

The crucial point in this corollary is to show that there

always exists a solution of the equation Lv = Cv, which grows

exponentially for x --f m r if r $2 IR. This can be done by looking

at the initial value problem with v prescribed as vo at xo E IR with xo sufficiently large. This problem can be reformulated as

X -ir 0 (Tv) (x,r) = I exp (x-Y)( ))-Q(y).v(y.c)dy

xO o i r ;

If xo is sufficiently large, then T is an operator on the

Banach space W = {v E C(IR) - - - I Im 5 I (x-x,) 2 lsup IV(XrC) le X&

1lvII < m } which has a norm small& than E with E > 0 as small W - - as desirable. It follows that: v = v u -ir, 0 vo(x,r) = exp ( (x - x ) (

+ (I-T)-lTvO with 0

i))vo. Consequently, we have

ZAKHAROV -SHABAT SYSTEM 157

u u

IIv-v 11 < 2~11; 11

creasing, then this is also the case for v.

and therefore, if vo is exponentially in- o w o w

Using corollary 5.3.1 it is possible to prove the following

result:

Lemma 5.3.1.

fi) If c o E IPu(L) n C+, t h e n c o is a simple z e r o of r-, i . e .

u

(ii) If c o E IPa(L) n C-, t h e n c o is a simple zero of r+, i . e . u d7 r+(co) = 0, b u t $ ( c o ) + 0 .

The proof of part (i) of this lemma proceeds by deriving a

contradiction from the hypothesis, that c0 E IPa(L) and .. ( g o ) = 0 for some n El". The idea is to - dLLr- r-(cO) = ... - -

drn perturb the system (5.3.1) slightly, i.e. we look at the

operator L , which is obtained by replacing the potentials r and q by r + E; and q + Eq with r,q E 0. This leads to a

corresponding function r-, which, if and are suitably

chosen, has n+l zero's in a neighbourhood of c o with diameter of the order E l'(n+l) with E .L 0. To each of these (n+l) zero's

there corresponds a point E IPa(LE) fl C+ with a 1-dimensional

eigenspace for E > 0, because of corollary 5.3.1. Taking the

limit E J- 0 and using the theory of continuity of finite

systems of eigenvalues, Kato, 1966, ch. IV, §3.4-5, it would

follow that there is an (n+l)-dimensional eigenspace

associated with c o E IPo(L) . This would contradict the corollary 5.3.1 and thus part (i) has been demonstrated.

It is left to the reader to fill in the details of this

reasoning and give an analogous proof of part (ii).

E

u - - E

In addition it is possible to derive explicit expressions for dr- dr - ( C 0 ) ,$(co) in a way analoqous to the case of the Schr6-

dinger equation (lemma 4.3.4). The reader may verify (or consult

Lamb,1980) that the following results hold:

dc


x,5 (x,CO)d.x O '2

Another interesting question is whether the number of discrete

eigenvalues is finite. The following result is an affirmative

answer to this question, but under a rather severe restriction.

Lemma 5.3.2. If f o r a l l 5 EIR the scattering coefficients

Satisfy r-(C) f 0, r+(c) + 0 then I P a ( L ) is a finite s e t . -

It is left to the reader to show this by combining the asymp-

totics given in (5.2.11), the continuity as expressed in the

corollary to theorem 5.2.3, the fact that r- and r+ do not

have real zero's and the fact, that the zero's of an analytic

function cannot have a limit point inside the domain of ana-

lyticity. It should be remarked, that the supposition, that

r- and r+ do not have real zero's is exactly what we shall need

for the derivation of the inverse scattering integral

equations. There this assumption arises in a quite natural way.

It is an open question how to give an estimate of the number

of discrete eigenvalues in terms of the potentials q, r of

the Zakharov-Shabat system (compare (4.3.15) ! )

-

-

5.4. FOURIER TRANSFORM OF SOLUTIONS.

The Fourier transforms with respect to the second variable of

the solutions L and will play a key role in the derivation of

the Gel'fand-Levitan-Marchenko integral equation describing the

inverse scattering. Here we shall derive some of the properties

of these Fourier transformed solutions. We define:


For the notation of Fourier transform and its properties we refer to section 4.4. Using theorem 5.2.3 , (5.2.4) and (5.2.10) it is clear, that L-(l) E F(lR+L2) and - 1 2 L - ( o ) E F( IR+L2). Consequently J and J are well-defined ele-

2 ments of F ( IR+L2). However, it is possible to say a lot more

about the properties of J and J.

0 2 - -

2 2 Theorem 5.4. J and ? a r e e l e m e n t s o f C( IR+Lz) n C( IR+L1) w h i c h

can be i d e n t i f i e d w i t h v e c t o r s of f u n c t i o n s o f t h e f o l l o w i n g

t y p e

f o r s > 0

f o r s C 0

- ii(x,s) f o r s > o

f o r s < 0

J(x,s) =

(5.4.2) J(x,s) = {N:x’s)

( 0 )

- 2 2 -

The k e r n e l s N and N a r e e l e m e n t s of C o ( I R + ) =

0 = {w E c2( I R ~ [ o , ~ ) ) IlimIw(x,s) I = ( o ) ~

F u r t h e r we have t h a t t h e v a l u e s o f t h e s e k e r n e l s a t s = 0

a r e r e l a t e d t o t h e p o t e n t i a l s in t h e f o l l o w i n g way

(5.4.3.) N(x,O) =

S+m

(-:“‘”’ 41s (y) r (y) dy

fls (Y 1 r (Y 1 dY

X m

The proof of this theorem is more or less analogous to the

proof of theorem 4.4.

Let us sketch a few steps in the case of J. We first note that,

a s a consequence of (5.2.10-c):


with hL E C( IR+L2) 2 fI C( IR+L1). 2

Hence :

X

- 2 2 with hL E C( IR+L~) n c ( m+c0).

A

As a consequence of lemma 4.4.1 we have hL(x,s) = 0 for s < 0. Hence it remains to be shown that J E C(IR+L1).

Using the differential equation for L given in (5.1.3-c) and applying Fourier transform to both sides we find that in

distributional sense:

2

a a (ax - 2z)J1 = qJ2 + q 6 ( s )

It is not difficult to show that this implies that N1 and N2

have to satisfy in classical sense

a - qN2 a (5.4.5) (ax - 2=m1 -

a N2 ax - rN1 - -

N1 (x,o) = -!iq(x)

lim suplNl(x,s) I = lim suplN2(x,s) I < x+m s a x+m s a

. This leads us to the following integral equations:


m

The equation f o r N1 can be solved iteratively m

N1 - - c Njn) (5.4.7) n=O

Njo) (x,s) = -4q(x+#s)

Njn+l) (x,s) = #/q(x+#(s-n) 1 S m

I (rNin) 1 (y, rl 1 dydrl 0 x+ki (s-rl)

and by induction with respect to n we find that

m

with M(x) = / I Ir(y) I + Iq(y) I lay.

Moreover, one can show that the solution given in (5.4.7) is

unique in w = Iv E c ( IR+) sup sup1 v(x,s) I < m ) .

x>a SX

Hence the solution given in (5.4.7) coincides with N as found

from (5.4.4) and (5.4.1,2).

X

7

1

The reader may now verify with induction to n that

m 2n (5.4.9) /IN?) (x,t) Idt < M(x+#s) for s 2 0

S

From hereon the proof of theorem 5.4 is easily completed.

An immediate consequence of (5.4.2) is

Corollary to theorem 5.4. The s o l u t i o n s

r e p r e s e n t a t i o n s of t h e f o l l o w i n g form

(5.4.10) f i ) L(x,C) = (1) + / N(X,S

f i i ) L(x,C) = ( o ) + .f N(X,S

m 0

0 0 1 -

0

1 -

the following result.

N

L and L have F o u r i e r

eiCsds

-iCs e ds


The i n t e g r a Z s i n t h e s e r e p r e s e n t a t i o n s a r e a b s o l u t e l y

c o n v e r g e n t .

Finally a result analogous to lemma 4.2 holds

Lemma 5.1.

l i ) The e x p r e s s i o n g i v e n f o r L i n (5.4.10)(i) is v a l i d f o r a 1 2

5 E z+.

a l l 5 E E - . -

(ii) The e x p r e s s i o n g i v e n f o r L i n (5.4.10) (ii) i s v a l i d f o r

The proof of this lemma is analogous to the proof of lemma 4.2.

5.5. INVERSE SCATTERING.

In this section we shall derive the analogue of the Gel'fand-

Levitan-Marchenko integral equation (see section 4.5) for the

Zakharov-Shabat system. In order to do this we make the

following additional assumption:

(5.5.1) V c EIR: r-(c) + 0 and r+(c) Z 0

In this way we guarantee, that - and 7 do not have singularities on the real axis.

%

1 1

r- r+

As a consequence of this assumption the number of discrete

eigenvalues will be finite (see lemma 5.3.2). Let us denote

eigenvalues in C+ by ck, k = l,...,d. Hence: r-(Ck) = 0,

k = 1, ..., d. The eigenvalues in C- are denoted by ck, k = l,...,d and they satisfy r+(ck) = 0, k = l,...,d.

Let us introduce the following notation:

- - - -

r+ (5.5.2) ar = - 1 , b r = -

r- r-

The relation of these coefficients to those in Ablowitz, 1978


1 - 1 - b a

and i n the chapters 6,7 is as follows: ar = a, ar = br - --, u - L - u - br - -=. Now w e s h a l l u s e , t h a t br and br have no s i n g u l a r i t i e s

on t h e real a x i s because of ( 5 . 5 . 1 ) . a

I n combinat ion w i t h ( 5 . 2 . 9 ) , (5 .2 .11) t h i s y i e l d s t h a t b, and br are e lements of L 2 n L~ n co ( IR) . W e conclude , t h a t t h e f u n c t i o n s Bcont ,

- -

d e f i n e d by: Bcont

- - - F-’br Bcont (5.5.3) 1 - u - - -

Bcan t 475 Fbr

Furthermore w e i n t r o d u c e d

To d e r i v e t h e second e q u a l i t i e s i n (5.5.6) w e use ( 5 . 3 . 2 ) ; r’(ck), r;(ck) canno t be equa l t o z e r o , because o f lemma 5.3.1. F u r t h e r w e remark t h a t Ck and Ek deno te t h e same c o n s t a n t s as i n Ablowitz , 1978 and i n chapters 6 and 7 .

Next w e d e f i n e

u u

- % - c o n t + B d i s c r = Bcont + B d i s c r ( 5 . 5 . 7 ) B = B

- I t i s an amusing exercise t o show t h a t B and B a r e r e a l f u n c t i o n s , i f t h e p o t e n t i a l s are real (see ( 5 . 1 . 8 ) ) . F i n a l l y w e i n t r o d u c e t h e n o t a t i o n


- with N, N as in (5.4.2). With these preliminaries

it is possible to derive the following result.

Theorem 5.5. Suppose t h a t t h e c o n d i t i o n s (5.2,3,4) and (5.5.1)

a r e s a t i s f i e d . Then t h e m a t r i x of F o u r i e r k e r n e l s [N] d e f i n e d

i n (5.5.8) s a t i s f i e s a m a t r i x v e r s i o n o f t h e G e l ' f a n d - L e v i t a n -

Marchenko i n t e g r a l e q u a t i o n :

(5.5.9) [N] (x,s) + [ B I (~x+s) + (x,t).[B] (2x+s+t)cit = 0.

The v a r i a b l e x a c t s a s a p a r a m e t e r i n (5.5.9).

The c o e f f i c i e n t s i n t h i s i n t e g r a l e q u a t i o n a r e d e t e r m i n e d by t h e f o l l o w i n g s c a t t e r i n g d a t a : t h e e i g e n v a l u e s yk, T k I t h e

n o r m a l i z a t i o n c o e f f i c i e n t s Ck, Ck and t h e f u n c t i o n s br, br

a s s o c i a t e d t o t h e c o n t i n u o u s s p e c t r u m .

m

0

- - 5

Let us sketch the proof of this theorem.

A transcription of the second relation given in (5.1.5-a) in

terms of 'ii, L, L is -

-2iyx - - arR = L + rjrZe

Next we insert the expressions for L and given in (5.4.1)

1 -2icx +

- - (5.5.10) arR - = V% Fi'J + ;,(ole

Note that each term in this identity is in C( 3R+L2).

Hence it makes sense to operate with F2 on this identity. A calculation analogous to the one immediately following (4.5.5)

shows, that for s > 0


h: - 'iYk(2X+S) = -i I: Ck~(x,ck)e

( . . I k= 1

= - Bdiscr (2X+S)4) - ( . . . I

N

In the last calculation we used successively: ( . ) contour

integration over a semi circle in C- and Cauchy's residue

theorem; (. . ) corollary 5.3.1; ( . . .) (5.4.10) I (ii) and lemma

(5.1) I (ii).

The conclusion is that F2 operating on the identity (5.5.10) leads to

m N

N(x,s) + g(2x+s) (i) + I N(x,t)~(Zx+s+t)dt = 0 0

(5.5.12)

In an analogous way starting with the first relation in

(5.1.5-a) we find:

(5.5.13) ii(x,s) + B(2x+s) (1) + j N(xIt)B'(2x+s+t)dt

It is easy to check, that (5.5.12-13) imply (5.5.9).

o m 0

Theorem 5.5 opens the way to an inverse scattering theory.

Given the scattering data ckl cklCkl~klbr,~r one can calculate

[ B l and then solve (5.5.9). The potentials q and r are found from N1, N2 in an easy way:

(5.5.14) q(x) = -2N1(x10) , r(x) = -2N2(x,0)

as we saw in (5.4.3).

N

5

N

Sufficient conditions have been found under which (5.5.9)

possesses a unique solution for all x Ern, for example in the

case r = -q*, see Ablowitz et al, 1974.


Assuming as in (5.2,3,4) that the potentials are continuous,

it would be sufficient if we could solve (5.5.9) uniquely for

the parameter x in a dense subset of JR. However,

the problem to characterize a large class of potentials q,r

for which this is the case is still open.

One can imagine a situation in which (5.5.9) is not uniquely

solvable €or the parameter x in some interval I CIR, where the

non-uniqueness is such that on I we cannot uniquely recon-

struct the potentials r and q from the given scattering data.

However, our feeling is that this pathological situation is

highly non-generic.

CHAPTER 6

APPLICATIONS OF THE INVERSE

SCATTERING TRANSFORMATION

We take up now the development started in chapter 3. In a rough

outline (omitting all technical details and conditions) we have

at our disposal the following results:

Let L be a family of operators on some Hilbert space, parame-

trized by t. The spectrum of L is invariant with t if there

exists an operator B such that

aL at - = BL - LB

The eigenfunctions satisfying

(6.2) Lv = gv

evolve according to

(6.3) - _ av - Bv at

In applications of the method of inverse scattering trans-

formation one looks for pairs of operators B and L such that the equation (6.1) is some interesting nonlinear evolution

equation for functions u(x,t) which occur as 'potentials' in

the operator L. For the succesful application of the method two further ingredients are needed: the inverse scattering problem

for (6.2) must be solved so that the potentials u(x,t) can be

reconstructed from scattering data; from consideration of

(6.3) one must be able to determine the evolution of the

scattering data with t.

It should be noted that if one aims at solving some particular

equation for u(x,t), then the construction of a suitable pair

of operators B and L can hardly be considered as a product of

16 7


deductive analysis, but should rather be identified as a

discovery.

In this chapter we describe a series of such discoveries. We

do not attempt to give a complete survey of the literature

pertaining to the search for evolution equations that are

integrable by the method of inverse scattering transformation.

Our aim is merely to show how the method works. In most of the

chapter the scattering problem generating integrable evolution

equation is the generalized Zakharov-Shabat problem, studied

in Chapter 5.

6.1. THE NONLINEAR SCHRODINGER EQUATION.

We consider the generalized Zakharov-Shabat problem

qv2 + irvl = 0 avl ax - -

where q(x,t) and r(x,t) are potentials for which we want to

derive evolution equations such that the spectrum of the

Zakharov-Shabat problem will be invariant with t.

Let v1,v2 be the components of a vector v. In vector notation (6.1.1) reads:

(6.1.2) Lv + icv = 0

with

The operator L is in general not selfadjoint, so that the

theory of sections 3.1 and 3.2 does not apply. However, by the

extension given in section 3.5, we still have for isospectral

potentials the Lax condition

(6.1.4) Lt = BL - LB

APPLICATIONS 169

where B is an operator defining the evolution of the eigen-

functions, i.e.

(6.1.5) - av - - Bv at

In this section we proceed as in section 3 . 3 and construct B

as a differential operator.

It should be clear that

(6.1.6) Lt

r O t

We write

(6.1.7) B = ('11 '12)

B21 822

Working out the condition (6.1.4), whil be i in mind that

Bij are, at this stage, unspecified operators, one obtains the following set of relations:

a a - ax B 11 - B,, 5 - qB2, - B12r = 0

a a (6.1.8)

82, - B, , ax - 62,s - rB12 = 0

a a T5 812 + B12 ax - 9822 + B11q = qt

a a - ax B 21 + B21 ax - rBll + B 2 p = rt

(6.1.9)

We now take Bij to be second order differential operators, i.e.

L

with as yet unspecified coefficients B On the left hand

side of (6.1.8) one then has third order differential operators

of which all coefficients must vanish. Similarly, the left hand

ij


sides of (6.1.9) are third order differential operators which

must be equal to the multiplication operators given by the

right hand sides. VJorking out the consequences one obtains the

following impressive s e t of conditions:

APPLICATIONS 171

Let us commence the analysis by rewriting (6.1.13) in a trivial

way as follows:

B;;) = (6.1.18) I

Substituting (6.1.18) into (6.1.14) shows that

If we would choose B i i ) to be inequal to zero, then, from (6.1.12) , Bi:) would be linear functions of x which would introduce in the evolution equation for eigenfunctions an un-

desirable behaviour for large 1x1. We therefore take

AS a consequence of this choice the coefficients ~1:) are independent of x.

We now proceed by introducing (6.1.18) and (6.1.20) into

(6.1.15) and obtain

Finally, introducing (6.1.18) and (6.1.21) into (6.1.16) yields:

Integrating we obtain

All elements of the matrix operator B are now fully determined, -

with yet the freedom to choose the 81;) and the integration


'constants' c1,c2, which can be arbitrary functions of t, but

independent of x. Substituting these results in (6.1.17) we find the evolution

equation for the potentials q(x,t) and r (x,t) :

Now, in the original Zakharov-Shabat problem one has:

(6.1.25) r = T q

where 4 is the complex conjugate of. q. With this choice (6.1.24) become

-

(6.1.26)

Clearly, the two equations are compatible if and only if

( 2 ) - B:;)) = ia, a real 4(622 (6.1.27)

and

(6.1.28) c2 - c1 = iy, y real.

Taking y = 0 one obtains:

which is the nonlinear SchrSdinger equation first solved by

inverse scattering transformation in Zakharov and Shabat (19721,

(1973).

The procedure of constructing the solutions is further

analogous to the analysis of the K.d.V. equation: considering

the evolution equation for eigenfunctions for 1x1 -+ m one can

determine the evolution in time of the scattering data: one is

then left with the task of solving the generalized Gel'fand-

Levitan equations (see chapter 5). Closed-form solutions can

again only be obtained in the case that the reflection

APPLICATIONS 1 7 3

coefficient is zero and the integral equations have degenerate

kernels. One then finds the pure FI soliton solution, corres-

ponding to N discrete eigenvalues of the Zakharov-Shabat

scattering problem.

It turns out that if one takes r = +6, which corresponds with the minus sign in the right hand side of the nonlinear

SchrGdinger equation ( 6 . 1 . 2 9 ) , there are no discrete eigen-

values. This case is therefore not interesting, from the point

of view of the occurrence of solitons.

Zakharov and Shabat ( 1 9 7 2 ) , ( 1 9 7 3 ) have studied in detail the

pure N soliton solution of the equation

( 6 . 1 . 3 0 ) iqt = qxx + q l q I 2

In the simples case N = 1 one finds the elementary soliton

solution

where 5 and 0 are the real and the imaginary parts of the

eigenvalue, i.e.

( 6 . 1 . 3 2 ) x = 5 + i 0

The function O(x,t) is given by

( 6 . 1 . 3 3 ) O(x,t) = exp{-2i5~+4i(5~-rl~)t-iO)

The constants x, and 4 follow from initial conditions.

The soliton thus has the structure of modulated oscillating

waves, of which the envelope moves with the speed 45, and

decays (exponentially) for 1x1 -+ m.

If N > 1, and the eigenvalues all have different real parts, then for large time the solution decomposes into N distinct

solitons of <he structure ( 6 . 1 . 3 1 ) . The only effect of soliton

interactions are phase shifts. In the case of eigenvalues with

the same real parts more complicated structures are found.


6.2. ISOSPECTRAL POTENTIALS FOR NONDEGENERATE FIRST ORDER

SYSTEPlS USING AN ALTERNATIVE APPROACH

The analysis given in the preceding section does not take full

advantage of the special structure of the Zakharov-Shabat

problem. Ablowitz, Kaup, Newel1 and Segur (1973, 1974) have

noted that all derivatives with respect to x in the evolution

equations for eigenfunctions can be eliminated usinq equations

(6.1.1). One can therefore, at the outset, introduce for B a

matrix of which the elements are simply functions of x and t, and depend on the eigenvalue 5.

Taking this point of view, and proceeding along the lines of

section 3.5, we shall derive a new, and rather simple,

characterization of isospectral potentials.

Let v be a vector with components vl,...,v . We consider spectral problems

(6.2.1). Lv + gv = 0

which can be formulated as a problem for a first order vector

differential equation

n

(6.2.2) av ax - = Av, A = A. + cA,

where A. and A, are matrixes of which the elements are functions of x and t. In the case of the generalized Zakharov Shabat problem one has

(6.2.3) A. = [ '1 , A, = (-i 'r 0 ' 0 i.

Let us consider briefly a more rjeneral system

(6.2.4) I- aw = A w ax

where J again is a matrix. If J is invertible, then (6.2.4) can be transformed to (6.2.2). We call the system in this case

nondenegerate. In this section we restrict ourselves to non-

degenerate systems, but one should not conclude too hastily

APPLICATIONS 175

that degenerate systems are of no interest. We shall return to

this question later on in this chapter.

Let now 5 = X be a point of the spectrum of L for t = to, and

let $(XI be a corresponding eigenfunction. We let v(x,t) evolve

according to

- = Bv, v(x,to) = $(x) at (6.2.5)

where B is a matrix. The problem (6.2.5) is uniquely solvable

on the whole time axis if the elements of B, for each x, are

continuous functions of t.

We now define a function f (x,t) by

(6.2.6) a

f(x,t) = - ax v(x,t) - A(’)v(x,t)

and differentiating we obtain

V af - a av A ( ~ ) 2 - aA(’) at ax at at at - _ _ - - (6.2.7)

with the initial condition

(6.2.8) f (x,t,) = 0

We use (6.2.5) , which yields:

av aA (’) v + B - - A(‘)Bv - - af . aB at ax ax at - = - (6.2.9)

Finally usincr (6.2.6) to get rid of

aB (A) - ax

af Bf = (- + BA (6.2.10) - - at

Let now the matrix B be such that

av ax

A(’)B - at)V

- we obtain

aA (’1

Then, by unique solvability of

(6.2.12) - af - Bf = 0 , f(x,tO) = 0 at


we have

(6 .2 .13) f(x,t) = 0 vt.

Hence 5 = A belongs to the spectrum for all time, and v(x,t) is

a corresponding eigenfunction.

The spectrum of L is thus invariant with t if there exists a

matrix B satisfyinq (6 .2 .11) . The advantage of this charac-

terization of isospectral potentials is that one does not have

to construct an operator B, as in chapter 3, and in section

6.1, but just a matrix!

The characterization (6 .2 .11 ) , with a formal justification

which consists of postulatinq (6.2.5) as evolution equation for

eigenfunctions and cross differentiation of (6.2.2) and (6.2.5)

has been used in Ablowitz, Kaup, Newell, Segur (1973, 19741,

Ablowitz and Haberman ( 1 9 7 5 ) , Ablowitz’s (1978) survey, and

various other publications quoted therein.

We now apply the characterization (6 .2 .11) to the generalized

Zakharov-Shabat problem. In that case n = 2; A(’) is given by

and hence

aA(l) O qt

(rt 0 (6 .2 .15) - = at

Writing

2 1 822

(6.2.16)

and working out the condition (6.2.11) one gets the following

set of conditions for the f u n c t i o n s Bij:

(6.2.17) + r B 1 2 - 9 8 2 1 = 0 ax aB,,

APPLICATIONS 177

(6.2.19) - + 2ihB12 + qT(B11-B22) = qt

2iXf321 - r(Bll-5 22 1 = rt

ax

a * 2 1 ax (6.2.20) - -

6.3. SOME EVOLUTION EQUATIONS FOR ISOSPECTIZAL POTENTIALS BY

AD HOC PROCEDURES.

In this section se describe some of the results of Ablowitz,

Kaup, Newell, Segur (1973, 1974), AKNS for short.

To start with we observe, by inspection of equations (6.2.17)

(6.2.18) , that

(6.3.1) f322 = - b l 1 + c(t)

In the AKNS analysis c(t) is taken equal to zero. One is then

left with the following set of equations:

(6.3.2) + rB12 - qf321 = 0 aB,,

ax

+ 2iXB12 + 2qB11 = qt ax aa,, (6.3.3)

- - a’21 2iX@21 - 2rBll = rt ax (6.3.4)

Let us first consider solutions for the functions Bij which

are second degree polynomials in A , i.e.

The procedure of determining Bij (0) ,Bij (1) ,Bij ( 2

section 6.1, however somewhat simpler.

From (6.3.3) , (6.3.4) it follows immediate X terms) that

(6.3.5)

This result, in (6.3.2) shows that 6:;) is

3

( 2 ) = p= 0 * 1 2 21

is analogous to

y (supressing the

independent of x.


Next, from (6.3.3), (6.3.4), supressing the h2 terms, one gets:

Using this result again in (6.3.2) leads to the conclusion that

B!:) is independent of x. We take

(6.3.7)

Supressing in (6.3.3), (6.3.4) the h term now yields 1

which in turn, in (6.3.21, implies

(2) + c(t) ( 0 ) (6.3.8) Bll = +fqrBll

With AKNS we take c(t) equal to zero. What remains now from

(6.3.3) (6.3.4) are the evolution equations

As in section 6.3, taking r = T { , and 6 ; : ) = ia, a real, one

gets the non-linear Schrgdinger equation. With the special

choice a = 2 the equation reads

(6.3.11) iqt = qxx * 2 q M 2

The corresponding elements of the matrix B in the evolution

equation for eigenfunctions

av (6.3.12) at = Bv

are given by

APPLICATIONS 179

One can consider in similar way solutions of the equations

(6.3.2) to (6.3.4) which are polynomials in X of a degree

higher than 2.

For a polynomial of third degree AKNS have shown that for r = 1

(in which case the Zakharov-Shabat problem reduces to the

SchrGdinger equation) one obtains the Korteweg-de Vries

equation. On the other hand, with a third degree polynomial in

A and r = Tq one obtains, as evolution equation for isospectral

potentials, the modified KdV equation

(6.3.14) 2 qt 2 6q qx + qxxx = 0

Other interesting equations arise if one looks €or solutions

of (6.3.2) to (6.3.4) which are of the structure

(6.3.15) 1 1 B l l = 7 a(x,t) : B 1 2 = b(x,t) ; B,, = - c(x,t) x

Straightforward substitution yields as conditions:

(6.3.16) = -rb + qc ax

ax ax ab - -zqa ; - - (6.3.17) - - - 2ra

= 2ib (6.3.18) {qt

t r = -2ic

Consider first the case

(6.3.19) r = -q

One can than take b = c, and one is left with

ab -2qa aa (6.3.20) ax = 2qb , - = ax


(6.3.21) qt = 2ib

Eliminating b with the aid of (6.3.21) leads to

aa - ax -iqqt (6.3.22) - -

(6.3.23) qxt = -4iqa

If one takes now

i a = a cos u q = -#ux

(6.3.24)

then (6.3.22), (6.3.23) are both satisfied provided that the

function u satisfies

(6.3.25) u = sin u xt

This is the well-known Sine-Gordon equation in characteristic

coordinates.

By a similar procedure, starting with

(6.3.26) r = q

taking b = -c and chosing further

i (6.3.27) a = a cosh u, g = %ux

one ends up with the Sinh-Gordon equation

(6.3.28) uxt = sinh u

The interested reader should consult AKNS (1973, 1974) and

Ablowitz (1978) for further details.

Let us summarize here the results obtained so far. We have

considered the generalized Zakharov-Shabat problem

- - avl qv2 + i6vl = 0 ax

ax - - a"' + rvl + icvz = 0

Following AKNS (1973, 1974), by simple ad hoc procedures,

evolution equations for isospectral potentials are obtained as

APPLICATIONS

follows :

181

- r = Tq : the non-linear Schrbdinger equation

r = l : the Korteweg-de Vries equation

= Tq : the modified KdV equation

r = -q = -+ux : the Sine-Gordon equation

r = q = fux : the Sinh-Gordon equation.

In all these cases, the elements of the matrix B in the

evolution equation for eigenfunctions, for any point x of the domain, are determined by the values of the potential and its

derivatives at that point. Therefore, for potentials that

vanish for 1x1 + m, the evolution equation for eigenfunctions

defines the time-evolution of the scattering data in a way

entirely analogous to the results of GGKM analysis described

in chapter 2.

6.4. THE GENERAL AKNS EVOLUTION EQUATIONS

One would wish to derive now general evolution equations

gouverning isospectral potentials for the generalized Zakharov

Shabat problem, which would contain the results of the

preceding section as special cases. Such equations have been

deduced in AKNS (1974). We shall describe here the main line

of the reasoning.

The first important observation is that the equations (6.3.2),

(6.3.3), (6.3.4), which define the matrix elements in the

evolution equation for eigenfunctions, can explicitly be

'solved' in terms of squared eigenfunctions. This can be seen

as follows:

Consider eigenfunctions defined by


From the equations of Zakharov Shabat problem one has

2 + 2iXJ11 - 2 ~ $ , $ ~ = 0 a$; - ax

a$; 2 (6.4.3) { ax - 2iX14~ - 2r$1$2 = 0

Compare these equations with the homogeneous version of (6.3.2)

to (6.3.4) I i.e.

- - + - - a B 2 1 2iXf321 - 2rBll = 0 ax

Simple inspection shows that (6.4.4) are satisfied by:

- - 2 - 2 (6.4.5) B,, = $,$, I B 1 2 = -$+ B 2 1 = $2

By similar considerations one finds two other linearly in-

dependent solutions of (6.4.4), which are given by

- 2 -2 - - - (6.4.6) B,, = T1T2t B12 = -$p B 2 , = q2

- - - - (6.4.7) B,, = $J2 + T1$21 B,, = -$l$lt B,, = $2T2

The general solution to the equations (6.3.2) to (6.3.4) can

now be found by elementary procedure of variation of para-

meters.

APPLICATIONS 183

The next observation is that in cases treated in section 6.3,

:. -n the potentials r and q tend to zero for 1x1 + m, one has:

lim B l l = A ( X ) X'T m

lim P12 = lim B Z 1 = 0 X+Tm X'T w

(6.4.8)

furthermore A ( X ) is independent of the potentials.

AKNS impose (6.4.8) as boundary conditions for the general

solution of (6.3.2) to (6.3.4). This leads to the following

conditions: -

The details of the analysis which leads to (6.4.9) can be found

in AKNS (1974) and in Ablowitz (1978).

The trick now is to get rid of X in the conditions (6.4.9).

From (6.4.31, eliminating $1$2 one gets n

2 X

A$: = -I-- 1 + 2q I (q$g + r$l)dx') 2i ax -m

.2i ax

(6.4.40)

2 X XQ2 2 = -I- 1 - 2r I (q$; + rJll)dx')

-m

The above relations also hold when $1,$2 are replaced by F,,?,. Hence in operator notation

with d: the matrix operator

X X - - a + 2q I dx'r -2r I dx'r --m -m ax

2i X X - - a 2r I dx'q -m + 9 I dx'9 ax -m


It follows that, if A(A) is a polynomial in A, then

(6.4.13) A(A)@(i) = A(L)@(i)

In AKNS (1974) the relation (6.4.13) is assumed to hold in the

more general case in which A(A) is an arbitrary entire function.

This generalization is however not quite trivial, because d: is

an unbounded operator with a restricted domain of definition,

so that some consideration of the interpretation and the

validityof (6.4.13) is required. We do not attempt to clarify

the matters here because in most applications A(A) indeed is a

polynomial or a ratio of polynomials, which is a case that we

shall turn to shortly.

Using (6.4.13) in (6.4.9) we obtain .-

Some further operations, involving adjoint operators, permit

to deduce

(6.4.15)

with

(6.4.16)

m m

d + 2r I dx'q 2q $ dx'q X X

ax 1 c+ = -{ 2i m m - - a -2q I dx'r

X ax - 2r I dx'r

X

Thus , (6.4.15) is satisfied if

r

(6.4.17) ( ') + 2A(g+)(:) = 0

-% This is the general AKNS evolution equation. By a similar

procedure, if A (A) is a ratio of polpanidLs

(6.4.18) A(A) =

then the evolution equation reads

ill ( A )

qni

APPLICATIONS 185

The function A ( X ) has a direct interpretation in terms of the

dispersion relation for the linearized evolution equation. From

( 6 . 4 . 1 7 ) , ( 6 . 4 . 1 6 ) , dropping nonlinear terms, one has l a

i a

r + 2A(- -)r = 0

-qt + 2A (2i a,)q = 0

t 2 i ax ( 6 . 4 . 2 0 )

Consider waves

( 6 . 4 . 2 1 )

then

( 6 . 4 . 2 2 ) iwr(k) = - 2 A ( 7 )

and hence

r = exp i[kx - wr(k)tl

k

1 2 1 r ( 6 . 4 . 2 3 ) A ( X ) = - - w (2X)

Similarly with

( 6 . 4 . 2 4 ) q = exp i[kx - w (k)tl

one gets q

1 2i q ( 6 . 4 . 2 5 ) A ( X ) = - w (2A)

This means that the dispersion relations must obey

( 6 . 4 . 2 6 ) w (k) = -wr(k) q

One can verify, starting with appropriate dispersion relation,

that the general AKNS evolution equations indeed contain the

cases treated in section 6 . 1 , with the exception of the

Korteweg-de Vries equation. The reason that the K.d.V.

equation is not contained will be explained shortly. First,

as an excercise, we consider the case of the nonlinear

Schradinger equation. In that case the linearized equation

and the dispersion relation is w(k) = -It . reads iqt =

The equation ( 6 . 4 . 1 7 ) becomes

2 qxx


Straight forward computation shows that

(6.4.28) E+(J = +itLX) qX

Hence, in the next step m m a + 2r J dx'q 2q J dx'q r X X

m m

- _ - a 2q $ dx'r X

ax -2r $ dx'r X

Working this out one gets

r (6.4.29)

The nonlinear Schrbdinger equation follows when one takes

r = T q (the complex conjugate)

Another easy exercise leads to the modified Korteweq-de Vries

equation. In that case the linearized equation reads qt = qxxx

and the dispersion relation is w(k) = -k3. The equation

(6.4.17 becomes

Taking for example r = q one obtains

The results obtained so far hold under the assumption that the

potentials r and q tend to zero as 1x1 -t m. There are therefore

not applicable in the case that r = -1, and the Zakharov-Shabat

problem reduces to the SchrBdinger equation. In AKNS (1974),

Appendix 3 this case is treated separately, along lines similar

APPLICATIONS 187

to the preceding analysis. The result is the evolution equation

the function y ( . )

by

(6.4.33) .y(k2)

The reader should

de Vries equation

KdV equation.

is related to the linear dispersion relation

verify that in the case of the Korteweq-

y ( k 1 = -4k2 and (6.4.25) indeed becomes the 2

6.5. DEGENERATE FIRST ORDER SCATTERING SYSTEMS AND THE SINE-GORDON EQUATION

In section 6.3 we have found that the Sine-Gordon equation in

characteristic coordinates yields isospectral potentials for

the generalized Zakharov-Shabatproblem. This was first

established in AKNS (1973). However, the reader must have

noted that among the evolution equations for isospectral

potentials derived in this chapter we have not encountered the

Sine-Gordon equation in 'laboratory coordinates', i.e.

(6.5.1) U tt - uxx + sin u = 0

Zakharov, Takhtadzhyan and Faddeev (19751, ZTF' for short,

apparently not aware of AKNS (1973), consider in connection

with equation (6.5.1) the scattering problem

1 Jv, + AV + - HV - LV = 0 5

(6.5.2)

where v isa two-dimensional vector, and

o w J = (:, il) , A = i(w o)

(6.5.3)

, w = u + Ut X

eiu o

l6 o e H = L(


One can arrive at the problem (6.5.2) by considering first a

four-dimensional system

- avl +-iwv 1 + v4 = cv2 ax 4 1

ax - -

(6.5.4) - eiuvl = 5v3 16

1 .-iuV 16 1 = 5vq -

The system clearly is degenerate in the terminology

6.2.

of section

Eliminating the components v3 and v4 one obtains (6.5.21,

which, as a consequence of the deqeneracy of (6.5.4), is non-

linear in the spectral parameter 5 .

Another example of a degenerate first order scattering problem

associated with an interesting nonlinear evolution equation

arises in the work of Zakharov and Manakow (19731, and is

discussed in Haberman (1976).

In this section we devote some attention to the scattering

problem (6.5.2). Our main objective is to show that methods

of chapter 3 also work in a somewhat unusual situation in

which the scattering problem depends nonlinearly on the

spectral parameter. At the end of the section we shall describe

briefly the behaviour of solitons of the Sine-Gordon equation.

In ZTF (1975) the assertion is made that solutions u(x,t) of

(6.5.1) are isospectral Dotentials for (6.5.21, while for

any point 5 = A of the spectrum the corresponding eigenfunctions v evolve according to

(6.5.5) 2 x

- vt - vX - - JHv The reader should be warned that in ZTF (1975) the evolution

equation for eigenfunctions contains a disturbing printing

error (the factor J in the second term on the right hand side

APPLICATIONS 189

is missing) which makes a direct verification of the assertion

virtually impossible.

We shall now proceed to demonstrate the isospectral properties

of solutions u(x,t) of (6.5.1), using a slightly different and

more symmetric version of the Z T F problem.

We observe that J2 = -I, the identity matrix, and obtain from

(6.5.2), for any point of the spectrum 5 = A

J{ ( A - A ) + fflv av ax - = (6.5.6)

Next, eliminating vx from (6.5.5) we have, as evolution

equation for eigenfunctions

(6.5.7)

Let now 5 = X be a point of the spectrum for t = to and $(x) a

corresponding eigenfunction. We let v evolve according to (6.5.7) , with ( v ) ~ = ~ ~ = 9, and we study the function f(x,t), defined by

1 - av = JE ( A - X ) - 7 fflv at

1 f = - t J{(A-X) + 7 fflv ax (6.5.8)

Differentiating with respect to t we obtain

1 av f = - - - a av t J{ ( A - X ) + HIS t ax at

(6.5.9)

From (6.5.71, differentiating with respect to x, we have

1 av 1 (6.5.10) 'a - - av = J{(A-1) - H}z + J(Ax - Hxlv ax at

Using this result in (6.5.91, and furthermore eliminating vt and vx by (6.5.7) and (6.5.81, we arrive at


(6.5.11) - af - JI(A-A) - 7 Hlf = at

1 1 - - J{(A-A) + 7 HIJ{(A-A) - 7 HIv

HlH{(A-A) + 1 Hlv - J{(A-A) - + J (At-Ax) v + 7J 1 (Ht+Hx)v

Finally, working out the right hand side, we find:

(A-A

+ Hx

Ax +

1 - - A Hlf =

- 2 (AJH-HJA) IV 2 (JH-HI) I

Using definitions (6.5.3) one finds by straiqht forward com-

putation that

(6.5.13) Ht + Hx - 2(AJH-HJA) = 0

Furthermore:

At - Ax + 2(JH-HJ) = x(utt i - uxx + sin u) (6.5.14) 1 0

Hence, if u(x,t) satisfies the Sine-Gordon equation (6.5.1)

then the problem for f becomes

(6.5.15) a f - J{(A-A) - - 1 HIf I [f]t,tO = o A

- - at

Under the condition of unique solvability of the evolution

equation for eigenfunctions we have f(x,t) = 0, t to, which

means that 5 = A belongs to the spectrum for t 2 to, and v is a corresponding eigenfunction.

The direct and the inverse scattering problem associated with

(6.5.21, (6.5.3) has been studied in Zakharov, Takhtadzhyan

and Faddeev (1975), where an outline of the solution is given.

A s we have mentioned, the Sine-Gordon equation in characteris-

tic coordination was studied by inverse scattering trans-

formation in AKNS (1973) using the Zakharov-Shabat scattering

APPLICATIONS 191

problem. The simplification that arises when one passes from

'laboratory' to characteristic coordinates can also directly be

seen in the formulation of the ZTF problem given in (6.5.61,

(6.5.8). Introducing the transformation

- (6.5.16) x + t = x , x - t = t

one immediately finds

(6.5.17) - av - - 2J(A-A)v ax _ - av JHv A (6.5.18) - -

af

Taking (6.5.17) as the scattering problem, all unusual

features are removed.

Both in AKNS (1973) and in ZTF (1975) the construction of

solutions follows the familiar path: consideration of the

evolution equations for eigenfunctions for 1x1 + m define the

evolution in time of the scattering data; the generalized

Gelfand-Levitan integral equation can be solved in closed form

only in the case that the reflection coefficient is zero and

the kernels are therefore deqenerate.

The discrete eigenvalues of the scattering problem associated

with the Sine-Gordon equation can be shown to be either purely

imaginary or arise in pairs A , -1 ( T complex conjugate).

One purely imaginary eigenvalue X = iq produces an elementary

sol iton

(6.5.19)

u(x,t) = 4 tan-lfe- + e I

1 1 8 = (ll + (x - x,) + (rl - =It

These travelling waves are called 'kinks'.

A single pair XI = 5 + iq, X 2 = - 5 + irl produces an elementary soliton


where v = 2 + - 2 . This soliton is called a 'breather'. 21hl

In the formulas given above x and t are 'laboratory

coordinates . In the general case of N discrete eigenvalues the solution can be given in closed form with the help of suitable determinants,

and the decomposition for large time into elementary solitons

can subsequently be studied.

6.6. HIGHER ORDER SCATTERING SYSTEMS.

Let us look again at nondegenerate scattering system of the

s t ruc ture - av = (6.6.1) ax (Ao + cAl)v

where A. and A, are matrixes, and 5 the spectral parameter.

Several results in the literature show that interesting non-

linear evolution equations are associated by inverse scattering

transformation to problem (6.6.1) in which v is a vector of dimension higher than two. We quote here the following three

examples :

The Boussinesq equation

2 2 - = - a u a u + at2 ax 2

(6.6.2)

A generalization of the n

4 - a2 2 i a u 2 u + - -

ax ax4

nonlinear SchrBdinger equation

(6.6.3)

2 a%,

at ax2 au2 a u2

at ax

i - aul = -+ (olu112 + uIu21 )ul

2 2

i-- - 2+ (olu112 + lllu21 )u2

where (I and p are constants.

The equations of the so-called three wave problem

APPLICATIONS 193

a A 1= v1 ax a A 1 + o(v2-v3)A2x3 a t

(6.6.4) - - a A 2 - v2 ax aA2 + (v1-v3)A1A3 at

+ F( (Vl-V2)X1A2 aA3 - a A 3

v3 ax - - at

where V1,V2,V3,u,p are constants.

The equation (6.6.2) was studied in Zakharov (

(6.6.2) as a system

(6.6.5) 1 4 xx

u = axx , (Dt = u + u2 + - u

974). Rewriting

Zakharov discovered that the pair of operators

4 % 2 \ + (7) u

dx

(6.6.6)

B =

satisfies the Lax condition in the form

(6.6.7) - - :t - -~(BL-LB)

The system (6.6.3) was studied by Manakov (1974); the system

(6.6.4) in Zakharov and Manakov (1976). In both cases

appropriate pairs of operators L and B are given (in a rather

complicated form). and the task of verifying (a version of)

the Lax condition for isospectral potentials is left as an

exercise for the reader. Luckily, the reader can discharge him-

self from this computational exercise by studying Ablowitz

(1978).

While in the work of hfanakov and Zakharov the operator B is

given as a differential operator (thus following closely Lax's

original idea), Ablowitz starts with the scattering system in

the form (6.6.1) and constructs appropriate operators B as

matrixes, satisfying conditions derived in section 6.2. The


Interested reader should consult Ablowitz (1978), where he

will also find other examples of evolution equations amenable

to analysis by the inverse scattering transformation.

CHAPTER 7

PERTURBATIONS

7 . 1 INTRODUCTION AND GENERAL FORMULATION

The n o n l i n e a r e v o l u t i o n e q u a t i o n s s o l v a t l e by an i n v e r s e s c a t t e r i n g t r a n s f o r m a t i o n t h a t w e have cons ide red ( t h e Korte- weg-de V r i e s e q u a t i o n , t h e n o n l i n e a r Schrgdinger e q u a t i o n , e t c . ) do no t c o n s t i t u t e "exac t " d e s c r i p t i o n s of n a t u r a l phenomena. The e q u a t i o n s are ob ta ined by i n t r o d u c i n g v a r i o u s approxi - mat ions i n t h e more complete govern ing e q u a t i o n s . I n a ra t iondl procedure t h e p rocess o f approximat ion c o n s i s t s o f d ropp ing terms t h a t are small i n some s e n s e . Converse ly , improving a mathemat ica l model c a n be cons ide red as i n t r o d u c t i o n of p e r t u r b a t i o n s i n t h e e q u a t i o n s d e s c r i b i n g t h e o r i g i n a l model. For example, a p e r t u r b e d Korteweg-de V r i e s e q u a t i o n w i l l have t h e s t r u c t u r e

( 7 . 1 . 1 )

where E is a small parameter , w h i l e f ( u ) i s terms depending on u and i t s d e r i v a t i v e s ; f i n an e x p l i c i t way on t h e v a r i a b l e s x and t

ut - 6uuX + uxXx = Ef(U)

some group o f u ) may a l s o depend

L e t u s recal l t h a t i n t h e framework of i n v e r s e s c a t t e r i n g t r a n s f o r m a t i o n a l l t h a t r e a l l y matters is t h e knowledge of t i m e - e v o l u t i o n of t h e s c a t t e r i n g d a t a . Hence, i n a p e r t u r b e d s i t u a t i o n , t h e f i r s t and fundamental t a s k i s t o derive e v o l u t i o n e q u a t i o n s f o r t h e spectrum, t h e r e f l e c t i o n c o e f f i c i e n t and t h e no rma l i za t ion c o e f f i c i e n t s . Such e v o l u t i o n e q u a t i o n s have been g iven i n t h e l i t e r a t u r e i n Kaup(1976) , Karpman & Maslov (1977) and Kaup & N e w e l 1 (1978) . However, the d e r i v a t i o n is u s u a l l y o n l y o u t l i n e d , (a r easonab ly complete accoun t can be

195

1 9 6 W. ECKHAUS & A . VAN HARTEN

found i n Kaup ( 1 9 7 6 ) ) , w h i l e t h e r e a s o n i n g f o l l o w s a r a t h e r d e l i c a t e p a t h i n t h e complex p l a n e .

I n t h i s c h a p t e r w e u se an e n t i r e l y e l emen ta ry approach t o d e r i v e t h e e v o l u t i o n e q u a t i o n s f o r t h e s p e c t r a l d a t a i n t h e c a s e of t h e S c h r 6 d i n q e r ' s e q u a t i o n and i n t h e case of t h e Zakharov-Shabat moblem. Fur thermore , i n bo th c a s e s , w e o b t a i n f o r t h e e v o l u t i o n o f t h e n o r m a l i z a t i o n c o e f f i c i e n t s s i m p l e r r e s u l t s t han t h o s e quoted above. We u s e i n t h i s t h e s i m p l i f i c a t i o n s of t h e c o e f f i c i e n t s needed i n t h e Ge l ' f and- L e v i t a n e q u a t i o n s which have been s t a t e d i n lemma 4.3.4 and e q s . (5.3.2).

I n d e r i v i n g t h e e v o l u t i o n e q u a t i o n s f o r t h e s p e c t r a l d a t a w e o f t e n proceed h e u r i s t i c a l l y a s w e d i d p r e v i o u s l y i n t h e c a s e o f unper turbed e q u a t i o n s i n t h e f i r s t p a r t of s e c t i o n 2 . 3 . 3 . T h i s e n a b l e s u s t o b r i n g o u t c l e a r l y t h e u n d e r l y i n g i d e a s . The d e r i v a t i o n s can be made r i g o r o u s by r e a s o n i n g ana logous t o second p a r t of s e c t i o n 2 . 3 . 3 , b u t w e do n o t i n c l u d e t h i s demons t r a t ion h e r e . The whole s t a t u s of t h e p e r t u r b a t i o n t h e o r y i s a t t h e p r e s e n t d a t e r a t h e r h e u r i s t i c , as w i l l become clear i n s e c t i o n 7 . 4 , which i s devo ted t o p e r t u r b a t i o n a n a l y s i s . I t is t h e r e f o r e c o n s i s t e n t t o a v o i d i n t h i s c h a p t e r l e n g t h y b u t n o t e s s e n t i a l d e m o n s t r a t i o n s .

W e now proceed t o t h e f o r m u l a t i o n of t h e problem of p e r t u r b e d i n t e g r a b l e e v o l u t i o n e q u a t i o n s .

We c o n s i d e r a s c a t t e r i n g problem

( 7 . 1 . 2 ) ( L + C ) V = 0

where L i s an o p e r a t o r c o n t a i n i n g a p o t e n t i a l u ( x , t ) . I n t h e unper turbed s i t u a t i o n u is a n i s o s p e c t r a l p o t e n t i a l and t h e e i g e n f u n c t i o n s 9 s a t i s f y i n g (7 .1 .2) e v o l v e a c c o r d i n g t o 0

The p e r t u r b e d e q u a t i o n t h a t u s a t i s f i e s c a n be w r i t t e n , u s i n g Lax formal i sm i n t h e form

PERTURBATIONS 197

d L - + LB - BL=-cf a t ( 7 . 1 . 4 )

where f i s some g i v e n o p e r a t o r . I t should be no ted t h a t when u s a t i s f i e s t h e p e r t u r b e d e q u a t i o n ( 7 . 1 . 4 ) , w i t h E # 0 , t h e e i g e n f u n c t i o n s do n o t s a t i s f y anymore t h e e q u a t i o n ( 7 . 1 . 3 ) . I n f a c t , t h e s t r u c t u r e of t h e e v o l u t i o n e q u a t i o n f o r t h e e i g e n f u n c t i o n s can b e c o n s i d e r e d as t h e p r i n c i p l e unknown of t h e p e r t u r b e d problem. Concerning t h e spec t rum of L w e s h a l l assume t h a t t h e d i s c r e t e e i g e n v a l u e s occur i n c o n t i n u o u s l y d i f f e r e n t i a b l e f a m i l i e s and t h a t t h e con t inuous p a r t o f t h e spectrum is i n v a r i a n t w i t h t . When (7 .1 .2) i s t h e SchrEdinger e q u a t i o n or t h e Zakharov-Shabat s c a t t e r i n g problem, t h e n t h e i n v a r i a n c e o f t h e con t inuous spec t rum c a n be a s s u r e d by imposing s u i t a b l e decay c o n d i t i o n s on t h e p o t e n t i a l s f o r 1x1 .+ m .

F i n a l l y , w e s h a l l assume t h a t when d i f f e r e n t i a t i n g (7 .1 .2) w i t h r e s p e c t to t , t h e u s u a l p r o d u c t r u l e h o l d s .

Now l e t C = X ( t ) be a f ami ly o f e i g e n v a l u e s and $ ( x , t ) t h e co r re spond ing e i g e n f u n c t i o n s . D i f f e r e n t i a t i n g (7 .1 .2 ) w i t h r e s p e c t t o t w e have

(7 .1 .5)

W e proceed i n t w o t r i v i a l steps as follows : a d d i n g and s u b t r a c t i n g (L+X)B$ w e g e t

( 7 . 1 . 6 ) ($LB)$ aL + XB$ + (L+X)($t-B$) = -At$.

Next , adding and s u b t r a c t i n g BL$ w e o b t a i n

Using ( 7 . 1 . 2 ) and ( 7 . 1 . 4 ) w e a r r i v e a t t h e f i n a l formula

(7 .1 .8) ( L + h ) ($,-B+) = (Ef-Xt)$.

2 Turning t o t h e con t inuous spec t rum w e t a k e 5 = +k a r b i t r a r y b u t f i x e d and $k a co r re spond ing e i g e n f u n c t i o n . Repea t ing t h e r eason ing w e g e t


(L+k 2 )(at - BJI ) = Ef+k . k (7.1.9)

The equations (7.1.8) and (7.1.9) provide the framework for our

further analysis. (Similar formulas occur in the work of

Karpman & Maslov (1977)). To obtain the evolution equations

for eigenfunctions we thus have to solve non-homogeneous

versions of the equations defining the scattering problem,

imposing proper behaviour for 1x1 + m.

However, the exercise is n o t entirely trivial. On the other hand, confining the attention to the evolution of the

scattering data (which are the quantities that are of real

interest to us) we shall find that in order to achieve the

desired results it is not necessary to go all the way in

solving (7.1.8) and (7.1.9)

In all that follows we assume that f vanishes (with u)

sufficiently rapidly for 1x1 + m .

7.2 EVOLUTION OF THE SCATTERING DATA IN THE CASE OF THE

SCHRUDINGER EQUATION

We now specialize to the case of a perturbed K.d.V. equation.

Then

(7.2.1) L = d'

ax - -

2 u(x,t).

Furthermore, from chapter 2 we have, for any point of the

spectrum 5

- aJI - BJI = - aJI - 2(~+25) ax aJI + (uX-C)$ at at (7.2.2)

where c is an undetermined constant.

7.2.1 The discrete eigenvalues and the reflection coefficient.

I t is almost trivial to establish the equation defining the evolution of discrete eigenvalue I; = X(t). Multiplying the

equation (7.1.8) by the corresponding eigenfunction and

integrating over the real axis one finds that the left hand

side vanishes andthus obtains

PERTURBATIONS 199

m

I fq2dx (7.2.1.1) - _ - a x - E

at y -m

where

(7.2.1.2) y = I $ dx.

The result (7.2.1.1) holds for any self-adjoint operator L.

m 2

-m

It will require a bit more of labour to establish the

evolution equation for the reflection coefficient b(k). We

take in (7.2.2) 5 = k and abbreviate 2

"k - - (7.2.1.3) at B$k = R.

We further recall that the eigenfunctions JI, satisfy

+ beikX for x -+ m -ikx

-ikx e e (7.2.1.4)

'k ae for x -+ -m . We multiply the equation (7.1.9) by an eigenfunction qk and integrate between any two pointsx and xo. This yields :

Next let xo tend to -01. From (7.2.2) using (7.2.1.4) one

finds

We obtain

2 aR R - = E J f$kdx'. -m 'k ax - ax (7.2.1.7)

In the final step we let x tend to +m. Using (7.2.2), (7.2.1.4)

one finds

3 -ikx (7.2.1.8) R = (s - 4ik3b)eikx + 4ik e + '$k

and hence


3 - R - % 2ikig - 8ik b). J,k ax ax (7.2.1.9)

Taking the limit in (7.2.1.7) we deduce

3 E m 2 (7.2.1.10) at ab - 8ik b = 2ik I fJ,, dx. -m

7.2.2 The normalization coefficient. .. L Let A = -kn be a discrete eigenvalue. The corresponding

eigenfunction behaves as follows :

'ne

-k x for x -P +m

n

+knx (7.2.2.1) J,

Erie for x -f -a .

We are interested in the time-evolution of the coefficient cn.

The equation (7.1.8) takes the form

with

R = J,, - BJ,

Unexpectedly, a procedure analogous to the preceding section

fails here. Let us briefly outline how this happens.

Multiplying (7.2.2.2) by J, and integrating one gets

2 X $Rx - RJ,x = I (f-At)J, dx'. (7.2.2.4)

4

Analyzing further this expression for x +. +m one gets an

identity.

Of course, (7.2.2.4) can be integrated again using

(7.2.2.5) J,Rx - RJ,x = J, 2 d - ( - ) . R dx J,

PERTURBATIONS 201

This does lead to solution of the non-homogeneous Schrtjdinger

equation (7.2.2.21, however, the path crosses various subtle

difficulties.

The analysis is simpler and more transparent if we modify

the starting point as follows :

First we specify the normalization of the eigenfunction $ by

imposing in (7.2.2.1 )

5

(7.2.2.6) cn = 1.

Next we introduce a second linearly independent solution

O of the SchrGdinger equation through the usual formula

(7.2.2.7) Ox$ - O$x = W

and we specify

One easily establishes that

(7.2.2.9) n' w = -2k 1 D n - - - , - 'n

The product O$ will play an important role in the analysis.

We have

@~ *[-; for x + +- (7.2.2.10) for x + --m

We now muitiply the equation (7.2.2.2) by the function 0

and integrate between any two points x and xo. This produces

(7.2.2.11) (@Rx,-ROxl 1 = E: j fQqdx' - X X

0 x'=x X


When xo tends to -m, from (7.2.2.31, using (7.2.2.1) , (7.2.2.81, (7.2.2.9) , one finds

% 8kn 4 + - dkn + 2kn + 2 k n x dkn x0. (ORx I +Ox I I =xo dt (7.2.2.12)

The presence of the last term on the right hand side of

(7.2.2.12) may seem disturbing at the first sight. However, in

(7.2.2.11) the second integral on the right hand side also

does not have a limit when xo +. --. With this motivation we rewrite

X X (7.2.2.13) A t $ @$dxl = At $ (Q$-l)dx' + At(x-xO)

xO xO

Recalling that

dkn (7.2.2.14) A t = -2kn dt

we take in (7.2.2.11) the limit for x + -m and obtain

(7.2.2.15) QRx - ROx = E $ f@$dx' - I (Q$-l) dx' - X X

-03 At -03

- A x + 8 k 4 +-+2kn dkn t n dt

In the final step we study the behaviour of (7.2.2.15) for

x +. +m. For the left hand side one finds

dkn + 2kn - dt

4 8kn + - 1 dCn ORx - RQx 2k - - - n Cn dt (7.2.2.16)

Again, the presence of the last term on the riqht hand side of

(7.2.2.16) seems disturbing at the first sight; however, in

(7.2.2.15) the limit of the right hand side for x +. +m also

does not exist. Balancing the misbehaving functions we take the

limit and obtain C m

dt 2kn -m

- - dCn 8knCn 3 = "{E $fQ$dx-XtO} (7.2.2.17)

X lim { J (0$-l)dx1+2x~ x+m -m

PERTURBATIONS 2 0 3

The reader may find it interesting to compare this result with

the corresponding evolution equation given in Kaup & Newel1

( 1 9 7 8 ) .

7 . 3 . EVOLUTION OF THE SCATTERING DATA IN THE CASE OF THE

GENERALIZED ZAKHAROV-SHABAT PROBLEM.

It will be convenient to adapt the general setting given in

section 7 . 1 along the lines of the formalism of section 6.2.

We briefly recall this point of view:

We consider the scattering problem

( 7 . 3 . 1 ) 2 = Av ; A = A. + i<Al ax

with

( 7 . 3 . 2 ) - 1 0

v = (v') , A. =

v1

and let v evolve according to

av at - = Bv ( 7 . 3 . 3 )

where B is a matrix, depending on the spectral parameter <.

The functions r and q are isospectral potentials if

( 7 . 3 . 4 ) aB - = - + BA - AB

at ax

We now perturb the evolution equations ( 7 ' . 3 . 4 ) by writing

( 7 . 3 . 5 ) - _ - - aB + (BA-AB) + EF at ax

with


In terms of the two potentials r and q the perturbed equations

have the structure

(7.3.7)

Assume again that the discrete eigenvalues of (7.3.1) occur in

differentiable families and that the continuous spectrum is

invariant with t. Let 5 = X(t) be a discrete eigenvalue and JI a corresponding eigenfunction. Differentiating (7.3.1) we get

From this, in a few trivial steps analogous to section 7.2,

one gets

a (7.3.9) (ax - A ) (2 - BJI) = EFJI + iXtAIJI

Similarly, for an arbitrary but fixed point of the continuous

spectrum, with JI (k) a corresponding eigenfunction, one gets

(7.3.10)

We recall from chapter 6 (in particular section 6 . 4 ) the

following facts concerning the elements B of the matrix B: ij

(7.3.11) B22 = - h 1

(7.3.12) lim B,, = lim B,, = 0 x+ T w x* Tm

(7.3.13) lim 8,, = A(<) x+ Tm

with A ( < ) indepedent of the potentials q and r. The statements

(7.3.121, (7.3.13) are true under the condition that r and q

tend to zero for 1x1 * m.

When manipulating the equations (7.3.9) and (7.3.10) we shall

frequently use the following result:

PERTURBATIONS 2 0 5

Lemma 7 . 3 . 1 . C o n s i d e r any p a i r of o e c t o r s

@ 1 R1

2 R2 I R = ( 1

and t h e i n n e r p r o d u c t

a @.(ax - A I R

where A i s a s g i v e n i n ( 7 . 3 . 2 ) . I f one t a k e s

w i t h

The proof is elementary.

7 . 3 . 1 . The discrete eigenvalues and the reflection coefficient.

Let 5 = A = 5 + in be a discrete eigenvalue, and let a corresponding eigenfunction $ b-e defined by

bn(l)e 0 ixx

r for x + +- for x -* -03

( 7 . 3 . 1 . 1 ) ' *

In the equation ( 7 . 3 . 9 ) we abreviate

( 7 . 3 . 1 . 2 ) - - a' BJI = R . at

We multiply ( 7 . 3 . 1 0 ) by

( 7 . 3 . 1 . 3 ) Q = 0 .1

and integrate using lemma ( 7 . 3 . 1 ) . One easily sees that

( 7 . 3 . la. 4 ) lim 0.R = 0

and thus obtains

X+TW


W co

(7.3.1.5)

Working out this expression produces

iht I @.Alqdx = -El @.F$dx. -03 - W

Y = 1 q1q2dX. --m

Now let 5 = k (real) be any point of the continuous spectrum.

We define a corresponding eigenfunction J, ( k ) by

O ikx for x +. +-m 1 -ikx

a(k) (,)e +b(k) (,)e (7.3.1.7) $ ( k ) %

for x -+ --m . We use here a notation different from chapter 5 (see remarks

in section 5.11, but analogous to Xaup & Newel1 (19781, to

facilitate the comparison of results. In the equation (7.3.10)

we abbreviate

For the components R(k), Rik) we have 1

Using (7.3.11) to (7.3.13) and (7.3.1.7) one deduces that

for x +. --m

Rjk) = - A(k)e-ikx

Rik) 0

(7.3 .l. 10)

-ikx

for x + +-m

Rjk) (2 - A(k)a)e Rik) * (at + A(k)b)e

(7.3.1.11) ab ikx

We multiply the equation (7.3.1 ) by

PERTURBATIONS 2 0 7

and integrate using Lemma ( 7 . 3 . 1 ) . This produces

Taking next the limit for x + +m one finds

One deduces without difficulty a similar result when

considering, instead of ( 7 . 3 . 1 . 7 ) , a wave coming in from the

left .

7 . 3 . 2 The normalization coefficient.

We proceed parallel to section 7 . 2 . 2 . Given the eigenfunction

$ defined in ( 7 . 3 . 1 . 1 ) we introduce a second linearly

independent solution

follows

5 of the Zakharov-Shabat problem as

with

- ( 7 . 3 . 2 . 2 ) bnbn = 1.

- We define

$ ( 7 . 3 . 2 . 3 ) @=(3 and deduces

[A(X)+iXtxl for x -+ -m

>+A (X 1 bn+iXtxbn] for x + +m

We multiply the equation (7.3.9) by the function and

integrate between any points x and x

( 7 . 3 . 2 . 4 ) @ . R

This yields 0'


X

(7.3.2.5) [ @ R I 1 =x - [@Rl = E l @F$dx' - x=xo xO

Note that

This motivates as in rewriting

Taking now in (7.3.2.5

X (7.3.2.8) @ R = E I

-m

the limit for xo + -03 we get

X @F$dx' - iXt (T2$1+~1$2)+lIdx +

-05

In a final step we study in (7.3.2.8) the behaviour for

x * +m. Using again (7.3.2.4) and (7.3.2.6) one obtains

m

(7.3.2.9) -gn[> + 2A(X)bn] = E I@F$dxl + -m

The evolution equation for the normalization coefficient bn

thus takes the form m

(7.3.2.10) - dbn + 2A(X)bn = bn(-iXtB + E .f(fl$lTl+f2$2$2)dx dt -m

" * 0 = lim I[ (~2$1+~1$2)+lldx1-2x~

x+m -m

In order to obtain the coefficients Cn needed in the Gel'fand-

lreviran eyuarions use snouia De maae or cne resuit given (in a

slightly different notation) in eqs. (5.3.2) and (5.5.6).

PERTURBATIONS

7.4. PERTURBATION ANALYSIS

209

Let us summarize and recombine the principal elements of the

mathematical structure that we have established. To be speci-

fic we focus on the K.d.V. equation.

In the notation of section 2.8 the Gelfand-Levitan equation is

given by

(7.4.1) (I+Td) f3 + Tcf3 = -R

R = Rd + Rc

The function Rc and the operator Tc are zero when the reflect-

ion coefficient is zero. The operator (I+Td) is invertible

(an integral equation with a degenerate kernel), and it is not

difficult to see working along the lines of section 2.5 that

the inverse

(7.4.2) S = (I+Td)

2 can be expressed in terms of the discrete eigenvalues X = -kn

and the normalization coefficients Cn without specifying their

behaviour in time.

-1

The solution f3(y,x,t) ofthe Gel'fand-Levitan equation repro-

duces the potential of the SchrGdinger equation

(7.4.3) 2 - - [u(x,t)-clv = 0

dx 2

through the formula

(7.4.4) u(x,t) = -- a B(o+;x,t) ax

In the case of reflectionless potential the pure N-soliton

solution u,(x,t) can be written out explicitly without

specifying the behaviour of Kn and Cn with time.

One can further write (7.4.1) in the form


(7.4.5) B = 6, - STcB - Sac B, = -sad

which was the starting point of the perturbation theory of

section 2.8.

On the other hand, we have established that if u satisfies the

perturbed K.d.V. equation

(7.4.6) ut - ~ U U + uxXx = E~(u) X

then the scattering data evolve as follows:

(7.4.7)

(7.4.8)

(7.4.9)

The functions

equation defined in section 7.2 and 8 is a function given in

(7.2.2.17).

qk and @ are solutions of the Schradinger

We remark that in the present chapter the eigenfunctions

corresponding to discrete eigenvalues are normalized in a way

different from chapter 2 (see section 7.2.2). Therefore, when

using formulas from both chapters care should be taken to re-

adapt the definition of the coefficients Cn.

The evolution of the scattering data becomes independent of

the potential u in the limit case E = 0; however, for E # 0

the whole structure is coupled.

when undertaking a Perturbation analysis for small E one

immediately faces some serious difficulties. In what follows

we develop an heuristic reasoning and discuss some open

PERTURBATIONS

problems.

211

Let us first consider time-intervals

(7.4.10) O Q t G T

where T is an arbitrary number, independent of E . Assuming in

(7.4.7) to (7.4.9) that f,$,$k,9 are bounded with respect to

t one easily deduces that

(7.4.11) X(t) = X ( 0 ) + O ( E )

(7.4.12) b(k,t) = e 8ik3tib (k I 0)

8k:t (7.4.13) cn(t) = e ICn(0) + O ( E ) )

(Note again that the discrete eigenfunctions are normalized in

this chapter in a way different from chapter 2. This accounts

for a different numerical factor in the exponent in (7.4.131,

as compared to Theorem 2.3.1) . Although the scattering data appear to be only slightly per-

turbed on the time scale under consideration, one cannot yet

conclude that the same holds for the solution u of the

perturbed equation. It is conceivable that on the basis of the

estimates given above, or suitable modifications thereoff, an

analysis parallel to section 2.8 could be developed, however,

at the present date this is only a conjecture.

We now consider larger time-intervals

(7.4.14) O g ~ t g T

where T is an arbitrary number, indepedent of E .

The evolution of the scattering data will in general be

affected by the perturbations in a non-negligable way. However,

one can still envisage a perturbation procedure on the basis

of (7.4.5) if one can assume that the reflection coefficient

b(k,t) is small on the time intervals under consideration.

Such assumption underlies the work of Kaup (1976), Karpman &

Maslov (1977) , Kaup & Newell (1978).

212 W. ECKHAUS & A. VAN EIARTEN

To be specific we take

(7.4.15) b(k,O) = 0

and write (7.4.8) in the form

E 8ik3t .-8ik3t' m 2 I fQk dxdtl (7.4.16) b(k,t) = 2ik e

-m 0

We now assume that in the integration with respect to t no

secular terms will occur. With this as a starting point one

can embark on the following procedure:

Step I. Take as a first approximation for u the pure N-soliton

solution uN in which the time behaviour of the discrete eigen-

values and the normalization coefficients is unspecified. Take

for the eigenfunctions in the first approximation the eigen-

functions corresponding to the pure-N-soliton solution.

Step 11. Compute the right hand sides of (7.4.7) and (7.4.9)

and solve the resulting evolution equations for kn and Cn.

Step 111. Compute the right hand side of (7.4.8) and verify

the non-secularity assumption.

Step IV. (optional) Proceed to higher approximations on the

basis of (7.4.5). One can also compute the contribution to u

by production of reflection coefficient b(k,t) from the

formula

N 2 z m kb* (kl j(dk (7.4.17) u = -4 Z kn$n + - n= 1 *i -m

(see section 2.5).

Final step. Demonstrate that one has indeed constructed an

approximation of U.

Applications of the procedure outlined above can be found in

Kaup (1976), Karpman & Maslov (19771, Kaup & Newell (1978) , where various perturbation problems are treated, related both

PERTURBATIONS 213

to the SchrSdinger equation and the Zakharov-Shabat. scattering

problem. The 'Final step' has, at the present date, not been

attempted. Another open question is a possible creation of

discrete.eigenva1ues that are not in the spectrum for t = 0.

In order to give an impression of the applications

we conclude this section with some computations that are

meant as an illustration of the steps I, I1 and 111.

We shall investigate, within the heuristic procedure, the

effect of the perturbations on the pure one-soliton solution.

(7.4.18) u(z,t) = -2kn 2 sech2 knz

- x = x - $(t)

A corresponding eigenfunction I) is given by

(7.4.19) I) = kn sech knz

The constant c1 follows from normalization

-knX -kn@ (7.4.20) lim e I) = a 2 n kne = 1

X+-m

which yields

Similarly, considering the behaviour for x + +m, we can

express the normalization constant C, in terms of 4 and obtain

(7.4.22) cn = e

We now consider the evolution of the eigenvalue from (7.4.7).

Straight-forward computation produces

2kn$

\

m dx dk

dt -0) cosh2 k,(x-@) " = - t e s f (7.4.23)

214 W. EKCHAUS & A . VAN HARTEN

Take for example

(7.4.24 ) f = uu

with u a constant. Equation (7.4.23) then reduces to

akn 2 - = - (7.4.25) dt 3 Eukn

The solution is

2 -u'Et 3 (7.4.26) kn(t) = kn(0)e

We now turn to the equation defining the evolution of the

normalization coefficient Cn (7.4.9). By a. straightforward

calculation one finds

(7.4.27) @$ = --{x a - tanh knzl

ax

and from (7.2.2.17)

(7.4.28) e = 24,

Using (7.4.22) we obtain

The expression naturally leads to the decomposition

(7.4.30)

with

(7.4.31)

4, = G1 + 4,2

4,,(t) = 4 1 kn(t')dt' + x 0 t 2

0 m

dt 4ki -m J f@+dx d4,2 E - = - (7.4.32)

If for example f is again given by (7.4.24) then

m

(7.4.33) J €@$ax = 0 -m

In this case, using (7.4.26), we obtain

PERTURBAT IONS 215

4 -0Et 2 1 3 - 11 + xo (7.3.34) 6(t) = 3kn(0) -[e U E

Clearly, the effect of the perturbations on the position of

the soliton is for large time quite considerable.

We finally study the evolution of the reflection coefficient

from (7.4.8). One can verify that with the potential (7.4.18)

the eigenfunction corresponding to the continuous spectrum is

given by

I (7.4.35) q k = e {I-- kn+ik cash kn(x-$)

-kn (x-0) - ikx kn e

(A nice method to construct eigenfunctionswhen the potential

is a pure N-soliton solution is given in Zakharov-Shabat

(1972) . )

The appearance of secular terms in (7.4.16) is highly unlikely,

unless f depends on t in an explicit way such that resonance

with other oscillatory factors in the integrant occurs. If f

depends only on u and its derivations, then we can write

-k n 3 2dx e k m -

- 2 i k ~ { ~ n (7.4.37) g(kn) = I f e -m kn+ik , Cosh knz

and (7.4.16) becomes

with w(t') = 2[4k2t'+4(t') 1

For a rough idea about the behaviour suppose that g(kn) is

differentiable with respect to kn. This is the case if one

considers for example (7.4.24). Integration by parts, using


some of the preceding results, then yields

2 (7.4.39) b(k,t) = -- E e 8ik't -2ik[4k t+$ (t) lgkn(t),- (2ikI2 4(k +kn) 2 2 Ie

-2ikxo 2 - e g[kn(0) I 1 + O ( E t)

Clearly, we indeed have

(7.4.40) b(k,t) = O ( E ) for 0 Q ct Q T

which assures consistence of the perturbation procedure. The

result is however only valid for k # 0, and in (7.4.39) the

quantity kb(k,t) develops a singularity as k + 0. To investi-

gate this phenomenon we return to (7.4.38) and compute

ikb(k,t) for k + 0, taking again as example (7.4.24). It is

easy to see that, for k = 0

(7.4.41) g(kh) = uknG

where s is a constant. Using (7.4.26) we find (7.4.42)

+€t - 11 3

[ikb(k,t) lk=O = 7 kn(0) [e

It follows that the behaviour of ikb(k,t) is non uniform when

both k and E tend to zero, and a more refined investigation is

required if one wishes to compute an approximation of u from

(7.4.17). One can expect that the production of non-zero

reflection coefficient which for large times is of order

unity in some small neighbourhood of k = 0 will result in a

contribution to U which is approximately constant (over some

interval). This phenomenon is called creation of a 'shelf' in

Kaup & Newel1 (1978).

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Documents

The Inverse Scattering Transformation and The Theory of Solitons: An Introduction