Physica A IStI (lYY2) 156170

North-Holland

Linearization transformations for non-linear dynamical systems: Hilbert space approach

1. Introduction

In the study of non-linear dynamical systems the discovery of exact solutions

is of great theoretical and practical importance. In rcccnt years remarkable

progress has been made in this direction. Let recall only the development of

the theory of soliton equations. The simplest strategy for the integration of :I

non-lincnr evolution equation is to try to linearize it. that is to construct an

explicit transformation mapping the non-linear equation to a linear one. The

only method known to the author for constructing the linearization mapping in

:I systematic manner is the Flato approach [ 1, 21. which is based on the theory

of non-linear representations of Lie groups. The significance of the Flato

technique for the theory of integrable dynamical systems might be far reaching.

but its USC as ;I practical tool for the linearization of the concrete non-linear

equations appears limited.

The purpose of this work is to introduce a method for finding the lincariza-

tion mapping with the use of the Hilbert space approach developed by the

author (3. 31. Namely. it is shown that the problem of constructing the

linearization mapping for non-linear dynamical systems with both a finite and

a11 infinite number of degrees of freedom can be reduced to the solution of a

linear equation in Hilbert space. The theory is illustrated by an example of the

Riccati system of the projective type and by the Burgers equation.

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K. Kowulski I Lirlearization trunsformations for dynamical systems IS7

2. Linearization transformations for ordinary differential equations

We begin with a brief account of the Hilbert space description of non-linear dynamical systems with a finite number of degrees of freedom [3], which is used in the construction of the linearization mapping. Consider the following analytic system of ordinary differential equations:

dxldt = F(x), 40) = XI, 3 (1)

where F: R/‘ + RX is analytic in x. Let Ix) be a normalized coherent state (see appendix A), where x satisfies

(1). We define the vectors Ix, t) such that

lx, t) = exp[ 4(x’ - xf)] Ix) . (2)

These vectors obey the following linear evolution equation in Hilbert space:

&t)=Mlx,t). ~x,O)=~xO),

where the boson operator M is of the form

M = u.’ -F(a) .

Here

a’ - F(a) = ,i, u; F,(a) .

Taking into account (2) we find that the following eigenvalue equation holds

true :

+,,3 t> = x(-q,, t) lx,,, t> 7 (4)

where Ix,,, t) is the solution of (3) and x(x,,, t) is the solution of (1). It thus

appears that the non-linear dynamical system (1) can be reduced to the linear

Schriidinger-like equation (3). It should be noted that the treatment can be

immediately generalized to include non-autonomous complex analytic systems

[31. We now discuss the transformation of variables within the Hilbert space

approach. Consider the following mapping:

x’ = 4(x) 7 (5)

where x satisfies (1) and C#J is analytic in X. It follows from (2) that under (5)

the “Hamiltonian” M transforms as

M’ = a’ *[4(a), Ml. (6)

Now let L: R/‘+ R/’ bc a linear operator. By virtue of (6) the linearization

mapping should satisfy

[+(a,. Ml = L+(u). (7)

On taking the Hermitian conjugate of (7) and using (A.15) we arrive at the

following system of linear equations in Hilbert space:

M'I&) = i &I$) . i-l...../\. (8) i 1

where 1~~5,) = 4,(a’) IO) and L,, is the matrix corresponding to the operator L.

Evidently. the linearization mapping C$ can bc recovered from solutions to

(8) by

4,(x) = (xJ4,) exp( 4-r’). i=l....,X. (9)

WC have thus shown that the problem of finding the linearization transform-

ation for non-linear systems of ordinary differential quations ( 1 ) can bc

reduced to the solution of the system of linear equations in Hilbert space (8).

Exwnplc 1. Consider the Riccati system of the projcctivc type 15-71

dxldr = Lx + (c . x)x , x(0) = .r, . (I())

where L: R“ + R” is a linear operator. The “Hamiltonian” M corresponding to

(IO) is

M=u'-La+ NC-U. (11)

where N=u’* a is the total number operator.

The following symmetry of the “Hamiltonian” can be easily obtained:

[M. N(l + L” ‘c.u)l=O. (12)

whcrc L: designates the transpose of L.

K. Kowalski I Linearization transformations for dynamical systems 159

Using (7), (12) and the Jacobi identity we find

[4;(a), N(l+ L-&z)] = 4;(a), i=l,...,k. (13)

Hence

(1 + L-‘c * aS)Nlc$;) = I&) ) i=l,...,k, (14)

which leads to

(N-l)(l+e-‘c..+)]~;)=o, i=l,...,k. (15)

Consider eq. (15). Since there is no non-trivial closed subspace of the Hilbert space, where the Bose operators act, which is invariant under their action, the only possibility left is

w- l)l$i) =o, i=l,...,k, (16)

where Itii) = (1+ L”-‘c*at)]&) and ]t+$) =0 only for I&) =O. The obvious solution of (16) is

I@;)=IO,...,O,l,,O ,..., O)=a~\O), i=l,..., k. (17)

Therefore, the solution to (15) can be written as

14;) = 4 1+ L”-‘c.a+

IO), i=l,..., k. (18)

Making use of (9) we finally find that the linearization transformation for the Riccati system (10) is of the form

4(x) = x 1+L-‘C.X. (19)

The transformation (19) for k = 2 was originally derived by Flato et al. [l] who used their group-theoretical approach. Based on the observations of Flato et al. the transformation (19) with arbitrary k was guessed by Reid (81. On taking the inverse transformation of (19) we obtain the following relation:

x’ x=

l- L”-‘c.x’ ’ (20)

where x’ = C+(X) is the solution of the linear part of the system (10) subJect to

the initial data

where F = 1 + L ‘c * x,,.

These initial data arc only non-singular t’or such x,, that F # 0. The singulari-

ty of the initial conditions can be easily rcmovcd by multiplying the numerator

and denominator of (20) by the factor F [9]. It follows that

x(x,,. I) = A(x,,. [)

I - r: ‘c.[A(x,,. l)-x,,] ’

where X(X,,. t) is the solution of (10) and ,4(x,,. I) is the solution of the linear

part of the system ( 10) subject to the initial data x,,.

Therefore (IO) can be cast into its linear part for arbitrary initial conditions.

The solution (22) was originally derived in ref. [9].

3. Linearization transformations for partial differential equations

WC now discuss the cast with an infinite number of degrees of freedom.

Before describing the method for linearization we first briefly sketch the

Hilbcrt space description of non-linear partial differential equations [Jl. Con-

sider the equation

d,u(x-. t) = F(u, D”u) , ff(S, 0) = u,,(x) . (23)

where u: R‘ x R+R, DC( = id”/d.~f’ . ilxf\, I,@/ = Xi , /3,. F is analytic in II.

D”a and u,, E Li(R‘. d‘x) (real Hilbert space of square-integrable functions).

Let us assume that the solution II of (23) is square integrable. Proceeding

analogously as in the cast with ;I finite number of degrees of freedom WC

introduce the vectors of the form

(24)

where II satisfies (23) and 1~) is a normalized functional coherent state (set’

appendix B).

The vectors (23) fulfil the following linear evolution equation in Hilbert

space:

K. Kowulski I Linearizution trunsformations for dynamical systems 161

(25)

where the boson operator M is given by

M = I

d”x a+(x) F(a(x), D”a(x))

Here u.‘(x) and u(x) are the standard Bose field operators. Let ]u,), t) designate

the solution of (25) and let u[u,]x, t] be the solution of (23) (the square

brackets denote the functional dependence of u on u,)). An immediate con-

sequence of (24) is the following eigenvalue equation:

Thus, it turns out that the non-linear equation (23) can be reduced to the linear

abstract Schrodinger-like equation in the Hilbert space (25). We note that the

treatment can be immediately extended to the case of complex multidimension-

al systems of partial differential equations (23) with a right-hand side depen-

dent on X, t [4]. We should also mention that the restriction imposed on initial

data is not too serious. In fact, the approach was shown to work also in the case

when the initial data were not square integrable [4]. On the other hand, the

requirement that the solutions are square integrable at any time is satisfied for a

large class of equations including the Korteweg-de Vries equation, the Burgers

equation (see example below), the non-linear Schrodinger equation and the

Kadomtsev-Petviashvili equation.

We now study the transformation of variables via the Hilbert space formal-

ism. By switching over to new variables

u’ = $[ulx] 1 (27)

where C#J is analytic in u, the “Hamiltonian” takes the form

M' = I

d”x u'(x)[c#+z\x], M]. (28)

Suppose that L is a linear differential operator. As in the case with a finite

number of degrees of freedom we find that the linearization mapping satisfies

the following commutation relation:

k#4+1~ Ml = Wbkl . (29)

Hence we obtain the following linear differential equation in Hilbert space

where 14(x)) = &[a’lx]lO) (see eq. (B.13)).

The solution of (30) is linked to the linearization transformation by

where 114) is a normalized functional coherent state.

It thus appears that the problem of determining the linearization transforma-

tion for non-linear partial differential equations can be reduced to the solution

of the abstract linear differential equation in Hilbert space.

Example 2. Consider the Burgers equation

d,ll=vi(U-Lla,u. u(x, 0) = u,,(x) E L;(R, dx) (32)

As we have mentioned earlier the solutions to (32) are square integrable at any

time. Indeed. we have

(33)

Therefore, the Hilbert space approach described above can be applied to the

Burgers equation. The conjugation of the “Hamiltonian” corresponding to

(32) is

M’zY I

dx (U’(X))“U(X) - I

dx (1’(X) (U’(S))‘U(X)

Now let L = v df. The equation (30) then takes the form

(31)

M’Ic#+)) = v &b(x)) (35)

Note that the solution of (35) is determined up to the additive factor

(ax + b)lO), where u, h are arbitrary constants. On writing (35) in the

coordinate representation (see appendix B) we arrive at the following linear

hierarchy equations:

“f,cp,(X;X,)=i9f~,(X;X,). (36a)

K. Kowalski I Linearization transformations for dynamical systems 163

where $n(x; x, , . . . , x,) = (x,, . . . , x,lq5(x)) and the reversed hat over x, denotes that this variable should be omitted from the set {x,, . . . , x,,+,}.

Performing the Fourier transformation we obtain from (36a), (36b) the system of linear algebraic equations of the form

(374

( ,I + I

= v k2- 2 k: &+,(k;k ,,... ,k,,+I), nil. r=l 1

Using the identities

(k2 - k; - k;)-‘6(k + k, + k2) = & 6(k + k, + k2), I 7

W’b)

(384

the following solution of (37a), (37b) can be easily found:

&(k; k,, . . , k,) = (-2: i)” (5 ilaCk + :, kj) (39)

On taking the inverse Fourier transformation of (39) and making use of

I+(x)) = 10) + 2 $ j- dxc, . . . dx, &,(x; x,, . . . , x,)jx,, . . , , x,) , (40) n=l

together with (B.3) we arrive at a solution of (35) such that

where c is an arbitrary constant.

Using (31 ) WC finally obtain the desired linearization transformation for the

Burgers equation

The inverse transformatiori

where (b satisties the linear

is of the form

part of (33) sub,jcct to the initial data

We have thus rediscovered the famous Hopf-Cole tr~insformation. which

reduces solutions of the Burgers equation to solutions of the heat equation

(note that (32) and (43) obtained with the help of the actual treatment hold

true regardless of the square integrability of li,,).

4. Discussion

Using the Hilbert space formalism developed by the author (see the book

[IO]) a new method is introduced in this work for the linearization of non-linear

dynamical systems with both a finite and an infnitc number of degrees of

freedom by reducing the problem to a linear equation in Hilbert space. The

linearization algorithm presented herein can be treated as a generalization of

the classical method of variation of constants to the case of non-linear partial

differential equations. In fact, any procedure for the reduction of a non-linear

ordinary differential system to the solution of its linear part like that presented

in section 2 has to be equivalent to the method of variation of constants. This

observation can be illustrated easily by the example of the Riccati system ( 10)

with an invertible matrix L [c)]. Furthermore, the introduced method of

linearization of partial differential equations is the most natural generalization

of the algorithm used for the linearization of ordinary differential equations.

On the contrary. the standard method of variation of constants seems to fail in

the case of non-linear partial differential equations. As an example consider the

K. Kowulski I Linearizution truttsformations for dynamicul systems 165

Burgers equation

a,u = v a$ - u a,vu

On setting

l4 = exp(vt a’> u(x, I)

we arrive at the following equation:

a,u = _ 2 C-24 a:U &.+lu

l! L 3 I =o

which has a non-linear, infinite-order, time-dependent right-hand side. The

above equation appears to be more complicated than the original Burgers

equation. In other words, the classical method of variation of constants does

not work in the case of the Burgers equation. On the other hand, the

celebrated Hopf-Cole transformation reducing the solution of the Burgers

equation to the solution of its linear part is easily obtained within the Hilbert

space approach introduced in this paper. The simplicity and universality of the

linearization technique described herein indicate that it would be a useful tool

in the study of non-linear dynamical systems.

Acknowledgement

This work was supported by KBN grant 2 0903 9101.

Appendix A. Bose operators and coherent states

We first recall the basic properties of the occupation number representation.

The Bose creation (a’) and annihilation (a) operators, where a’ =

(a: , . . . , a:), a = (a,, . . , a,), satisfy the Heisenberg algebra

[a,, a:] = ',j , (A.la)

[a;, ai] = [at, a:] = 0, i, j=l,..., k. (A.lb)

We assume that there is no non-trivial closed subspace of the Hilbert space of

states 2 that is invariant under the action of the operators a and at. Suppose

that there exists a unique normalized vector IO) (vacuum vector) in Xsuch that

alo) = 0. (A.2)

The state vectors In). n E Z”, , where Z + is the set of non-negative integers, arc

defined as follows:

(A.3)

They are the eigenvectors common to the number operators N, = a,‘~,, i =

1.. . . k, that is

N/n) = nln) . (A.3)

These vectors form an orthogonal and complete set. We have

(A.5)

c IM4=~. (A.61 “EZ:

The vectors In) span the occupation number representation. The Bose

operators act on these vectors as follows:

u,(n) = filn - e,> 1 u,‘In) = j/-In + e,) . (A.7)

where e, = (0. . , 0, I,, 0, . . (I), i = 1, , k, are the unit vectors of R”.

We now summarize the basic properties of the coherent states. The coherent

states 12). where z E CA, are usually defined as the common eigenvectors of the

annihilation operators

alz) = zlz) (A.to

The normalized coherent states can also be defined as

lz) =exp(-41z12)exp(z.a’) IO), (A.9)

where (~1’ = C:_ , Iz,(~ Th e coherent states form the non-orthogonal set. We

have

(zlw) =exp[-l((z(‘+(wJ~-2z”.w>]. (A. 10)

where the asterisk designates the complex conjugation.

K. Kowalski I Linearization transformations for dynamical systems 167

The completeness condition for the coherent states can be written in the

following form:

I d/-M lz> (~1 = 1, (A.ll)

R=

where

k 1 dp(z) = n - d(Re zj) d(Im z,) .

r=l n

The matrix element describing the passage from the occupation number representation to the coherent states representation is

(4~) = (,fj $1 exp(-ilz12). I.

(A.12)

In particular, we have

(O(z) =exp(-$lz12). (A. 13)

Suppose now that 14) . 1s an arbitrary state. Using (A.6) and (A.12) we find

that the function +(z*) = (zl$> can be written as

4(z*) = $(z*> exp(- 4 1~1’) , (A.14)

where C&Z*) is an analytic function (entire function). Taking into account (A.8), (A.13) and (A.14) we obtain the following

abstract, basis-independent form of (A.14):

I+> = &a’Mo) . (A.15)

The relations (A.ll) and (A.14) taken together imply that

($14) = 1 Q-4) exp(-lz12) &*<z*> i%z*) . R2k

(A. 16)

We have thus shown that the abstract vectors can be represented by analytic functions. Such a representation is called the Bargmann representation. The Bose operators act in the Bargmann representation as follows:

ai(z*) = j$ i<z*> > at&z*) = z* c&z*> . (A.17)

Appendix B. Bose field operators and functional coherent states

We begin by recalling the basic properties of the coordinate rcprcscntation.

The Bose field operators obey the canonical commutation relations

[N(.Y). (I (x’)] = q.t- - x’) . (B.1)

[u(_v), (1(X’)] = [(I’(_\-). d(d)] = 0 . .\‘. .v’ E R’

As in the cast‘ with finite number of degrees of freedom we assume’ that there i4

no non-trivial closed suhspacc of the Hilbcrt space of state4 H that is invariant

under the action of the operators o(s) and a’(.~‘). Suppose now that there

exists in if a unique normalized vector IO) ( vxuuni vector) satisfying

~l(.Y)JO) = 0 . for cvcry .V E R‘ , (B.2)

The vectors I.\-, . . x,,) _ x, E R‘. tlcfincd ;IS follows:

).Y,, . x,,) = (i:] i(X)) )I(!) . / I

(B.3)

form the orthogonal and complete set. We have

(A,. . x,,/x; . . ..t-,:,) = 6 ,,,, I 2 f1 8(.rt .t((,,) . (B.4) ‘r / I

where u is ;I permutation of the set ( I, . , II).

CL L I, n! I d ‘x / d ‘.Y,, Is, , . x,, ) (x, . . . A-,! 1 = I (B.5)

The vectors (B.3) span the coordinate representation. The Bose field operators

act on the vectors I.\ ,, r ) in the following way: 1, ,,

rr(.u))_r , , , _Y,( ) = i 6(x ~ .t-, )/.u, , . if. . _\‘,j ) . , I

(B.6)

u’(x)Ix,. . ,x,,) = lx,, . . a,,.x) .

where the reversed hat over X, denotes that this variable should be omitted

from the set (s,, . , _I-,,). WC now sketch the basic properties of the functional coherent states. The

functional coherent states Jo), where u E L’(R’, d’x) (the complex Hilhert

K. Kowalski I Linearization transformations for dynamical systems 169

space of square-integrable functions) can be defined as the eigenvectors of the annihilation operators,

44lu) = 444 . (B.7)

The normalized functional coherent states can be defined in an equivalent way as

ILL) = exp(- i 1 d”x 1~1’) exp(/ d”x u(x)Q~(x)) IO) . VW

These states fulfil the following relations:

(ulu) = exp(- f 1 d‘x ()u(2 + Iu12 - 2u”u)) ,

i D’u ]u)(u\ = I,

02

(B.9)

(B.lO)

where 0 = Y’(R’) or 0 = g’(R”) (Y(R’) and g(R”) are the standard test function spaces in the theory of distributions), D’u = D(Re u) D(Im U) and exp(-] d‘x u’) Du, where u E La(R‘, d‘x) designates the Gaussian measure on iI.

The passage from the coordinate representation to the functional coherent states representation is given by

(x,, . . . , x,Ju) = (fi u(x,)) exp(- k 1 d”x 1~1~) . 1=I

(B.ll)

Now let I$) be an arbitrary state. Taking into account (B.5) and (B.11) we find that the functional (b[u*] = (~14) can be written as

(B.12)

where the functional &[u*] is analytic.

Proceeding analogously as in the case of the finite number of degrees of

freedom we obtain the abstract, basis-independent form of (B. 12)

14) = &z’]lO) . (B.13)

The following relation is an immediate consequence of (B. 10) and (B.12):

The representation (B.14) is the functional Bargmann representation. The

action of the Bose operators in this representation has the following form:

[,I@) &*I = u:yx) &:!:]

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