Volume 145, number 8,9 PHYSICS LETTERS A 30 April 1990

ON THE MODULATION OF SOLUTIONS TO SCALAR LAX EQUATIONS ACCORDING TO WHITHAM’S PROCEDURE

Ian MCINTOSH Department ofApplied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9E W, UK

Received 16 November 1989; revised manuscript received 5 March 1990; accepted for publication 6 March 1990 Communicated by A.P. Fordy

Flaschka, Forest and McLaughlin have demonstrated that Whitham’s modulation equations for generic “‘g-8ap” solutions of the Korteweg-de Vries equation can be expressed as the variation of certain meromorphic differentials over a family of Riemann surfaces. Here we derive the analogous result for an arbitrary scalar Lax equation.

1. Introduction

In recent years there has been a renewal of interest in Whitham’s method (see refs. [ 1,2] ) for describ- ing the modulation of periodic or quasi-periodic so- lutions of wave equations, particularly when these wave equations possess an infinite number of con- servation laws (for example, refs. [ 3-8 ] ) . In ref. [ 3 1, Flaschka, Forest and McLaughlin derived a concise formulation of the modulation equations for g-phase quasi-periodic solutions (belonging to the class of so- called g-gap or g-zone solutions) of the Korteweg-de Vries equation. The object of this note is to show that there is a simple way of determining the analogous formulation for an arbitrary scalar Lax equation. The key to this formulation is the elegant description of conservation laws for scalar Lax equations given by Wilson in ref. [ 91. We begin by recalling this description.

2. Conservation laws for scalar Lax equations

Recall that a scalar Lax equation is an equation for the variables uo, . . . . u,_ ,, which can be written in the form

L,=[P,L]. (1)

Here L=a”+‘+u,_,a”-‘+...+uo (where c?=c?/I~x) and P is also a differential operator whose coeffi-

cients are polynomials in the dependent variables and their x-derivatives. It is convenient to introduce the differential algebra 9 of all such differential poly- nomials in the variables uo, . . . . u,_ ,.

It is shown in ref. [ 91 that there exists a particular formal series in {,

x= 1 +x,r-I+... (2)

satisfying Lxtic= Cn+‘xexC, whose coefficients Xi are (homogeneous) differential polynomials belonging to a suitable extension 3 of Z@ i.e. the Xi are not sim- ply differential polynomials in the original variables, they involve some integrals (more details of this, which are unnecessary for our discussion, may be found in ref. [ 10 ] ) . An infinite sequence of conser- vation laws for eq. ( 1) (and indeed any equation in the same hierarchy) results from:

Proposition (3.11 of ref. [ 91). The coefficients Hk, Jk of the formal series

xXx-‘= F Hk{-k, X,X-‘= EJkc-k, I I

belong to a, hence the condition that alax and d/at commute implies the conservation laws

(Hk)r= (Jk)x-

The formal series x is used to construct the formal “Baker-Akhiezer” function p= exp (xc+ tC” )x which

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Volume 145, number 8,9 PHYSICS LETTERS A 30 April 1990

is the formal series solution of the “direct spectral problem” for ( 1);

Lw=r”+‘w, V*=PV, (3)

where the leading order term of P is 13~. Clearly the conservation laws above are contained in the equation

(Kw-‘)l=(w*w-‘), * (4)

This relationship between the conservation laws and the formal Baker-Akhiezer function is fundamental to what follows.

3. Whitham’s modulation equations

We will simply write down the modulation equa- tions directly from the conservation laws, without regard for their derivation (readers may consult refs. [ 2,4,7,11] for differing attitudes towards the deri- vation). First, it should be pointed out that the equa- tions presented here can only hope to describe the modulation of a generic g-phase quasi-periodic so- lution of ( 1 ), in the following sense. We fix a so- lution vector U(X, t; AI) = ( uo, . . . . u,_ , ) by fixing the values of the parameters A) This solution is chosen so that in some neighbourhood of u(x, t; Aj) in pa- rameter space (i.e. for suitably small perturbations of the AI) the solutions corresponding to this neigh- bourhood all have g independent phases in both x and t. This guarantees that U(X, t; Aj) is not a “res- onant” solution.

To modulate this g-phase wave, we suppose that the parameters Aj now depend upon variables X and T which, for our purposes, are independent of x and t. The equations governing the variation of the Aj are

= & (J/c[a(-% t; Aj) I> * (5)

Here Hk[ II (x, t; AI) ] means the function of (x, t; AI) obtained by evaluating the differential polynomial Hk at u (x, t; AI). The angled brackets denote the fol- lowing average. Since u is a g-phase quasi-periodic function in both x and t, so is Hk[u]. Therefore it can be viewed as a function defined almost every-

where on a certain g-dimensional torus Y, upon which both the x and t flows wind ergodically. The average in (5) refers to the average over this torus:

(fh[ul> -1

= (5

H/c(h(x, t), . . . . @&, t) )w )(I >

0 . .F r

(6)

Here the @i are coordinates on Y and o is a volume form. The fact that the x and t flows are both ergodic on this torus means that this average is identical to the “average over long time” in either x or t.

Now we notice that all the equations in (5 ) could be equally well written down using (4) after replac- ing the formal Baker-Akhiezer function by the so- lution of eqs. ( 3 ) corresponding to our solution u (x, t; Aj) of ( 1). The formula for this solution of (3) is (cf. ref. [ 12 1, formula (9.13 ), for example)

Z

V(X, t, Z; Aj) = exp U

(8,x+i&t) -k,x-k,t > PO

XQ(& t, Z;Aj) 9 (7)

where Q( x, t, z; Aj) is an almost everywhere nonzero g-phase quasi-periodic function of x and t, involving a ratio of &functions on a certain Riemann surface W (this Riemann surface is compact with genus g and depends smoothly upon the choice of parame- ters Aj). The construction of this function (7 ) is de- tailed in ref. [ 121, but we will only be interested in the exponential term. The integrals in this term in- volve two meromorphic differentials Q, and Sz, on R whose only poles lie at “the point at infinity” p, (this is a distinguished branch point of order n on L@);z-‘isalocalparameteraboutp, (say, ]z-‘I<1 with z-’ (p,) =O) and p. is an arbitrary base point. All we need to know here is that, if p. is different from p,, ~(x, t, z; Aj) converges for O< ]z-‘1 < 1 and has the series expansion exp( xz+ tz”‘) x [ 1 + 0 (z- ’ ) ] in this region. The implicit depen- dence of 9 on the parameters Aj means that Q,, .R,, kr, k,,, also depend upon these parameters.

When we replace the formal Baker-Akhiezer func- tion in (4) by (7), we obtain a single equation which contains all those in ( 5):

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Volume 145, number 8,9 PHYSICS LETTERS A 30 April 1990

PO

Since both x and t flows are ergodic on Y it is a sim- ple application of Stokes’ theorem to show that both (Q&-‘) and (Q&-l) vanish. Hence we are left with

From ref. [ 12 ] we learn that the constants k,, k,,, are chosen so that J&sli - ki has no constant term, there- fore the equations obtained by equating coefficients of zek in (9) are identical to those given by equating coefficients of d( zTk) in

(10)

When ( 1) is the KdV equation, this is precisely the concise formulation obtained in ref. [ 3 1. Note that the same expression has been derived by Krichever [ 71 as part of his scheme of modulation equations for all systems of KP-type. However, in ref. [ 7 ] the Whitham equations (5) are not taken for granted, instead we are led through a difficult argument to- wards the result.

Remark. It is perhaps worth a brief remark con- cerning what has been achieved. Firstly, notice that eq. (9) does not appear to be of much more use than the sequence of equations in ( 5 ), since powers of the local parameter z--I merely index this sequence. However, ( 10) is valid everywhere on the Riemann surface. Flaschka et al. [ 31 made good use of this

fact (in the case of the KdV equation, where m = 3, n= 1 and %? is hyperelliptic) by showing that the fi-

nite branch points (or more correctly, the finite

Weierstrass points) Ai of W are Riemann invariants,

advected with speed 8,(nj)/s2,(n,) i.e. {a,-

[sz,(ni)/a,(~i)]ax};li=O. In fact it is a straightfor- ward calculation to show that this result holds for arbitrary m, n= 1, W hyperelliptic i.e. when ( 1.1) is

the mth equation in the KdV hierarchy. When 9 is not hyperelliptic the explicit form of

the differentials 8,, a,,, is not known in general, so the problem of computing the Riemann invariants is non-trivial. The matter is further complicated if one is interested in the physical problem of the modu- lation of real-valued, bounded solutions. It is ap- parently an open problem to characterize, amongst Riemann surfaces which correspond to algebraic functions over [R, those for which the function Q(x, t, z; AI) normalizes to a nowhere zero real-valued function for real-valued x, t.

References

[ 1 ] G.B. Whitham, Proc. R. Sot. A 283 (1965) 238. [2] G.B. Whitham, Linear and non-linear waves (Wiley-

Interscience, New York, 1974). [ 3 ] H. Flaschka, M.G. Forest and D.W. McLaughlin, Commun.

Pure Appl. Math. 33 ( 1980) 739. [4] S.Yu. Dobrokhotov and V.P. Maslov, J. Sov. Math. 16

(1981) no. 6. [ 51 B.A. Dubrovin and S.P. Novikov, Sov. Math. Dokl. 27

(1983) 665. [6] S.P. Tsarev, Sov. Math. Dokl. 31 (1985) 488. [7] I.M. Krichever, Funct. Anal. Appl. 22 ( 1988) 200. [8]C.D.Levermore,Commun.P.D.E. 13 (1988) 495. [ 9 ] G. Wilson, Q. J. Math. Oxford 32 ( 198 1) 49 1.

[ lo] G. Wilson, Math. Proc. Cambridge Philos. Sot. 86 (1979) 31.

[ 1 I ] P.D. Lax and C.D. Levermore, Commun. Pure Appl. Math. 36 (1983) 253.

[12]G.B. SegalandG. Wilson,Publ.Math.I.H.E.S. 61 (1985).

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