Linear integral equations and nonlinear difference-difference equations

408

Nonlinear

science

abstracts

LINEAR INTEGRAL EQUATIONS AND NONLINEAR DIFFERENCE-DIFFERENCE EQUATIONS, G. R. W. Quispel, F. W. Nijhoff, H. W. Cape1 and J. Van der Linder, Instituut-Lorentz voor Theoretische Natuurkunde, Nieuwsteeg 18, 2311 SB Leiden, THE NETHERLANDS. In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of BZcklund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differentialdifference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference version and the differentialdifference version of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schr6dinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation. A SINGULAR INVERSE-SCATTERING APPROACH TO HYPERBOLIC SYSTEMS, C. Cercignani, Dipartimento di Matematica, Politecnico di Milano, Milan0 (ITALY). It is well-known that the nonlinear equations admitting an inverse Accordingly, any treatment of scattering approach describe dispersive phenomena. hyperbolic systems based on the inverse scattering method appears to be doomed to failure. However, the circumstance that such systems may, in particular cases, be the limiting forms of nonlinear dispersive wave equations suggest that a sort of singular version ofthe method should be applicable. In this paper the question is studied in detail, with particular refernce to the simple equation

(1)

Ut

+

uux

=

0

which is, quite clearly, a limiting case of the KdV equation. known general integral, given, in an implicit form, by

(2)

u

=

Eq. (1) has a well

g(x-ut)

where g is an arbitrary function (essentially the initial datum u(x,O) = u,(x)). This classical result will be ignored and found as a consequence. Rather, the starting point of the analysis will be the remark that Eq. (1) admits a Lax pair (L,B) defined by L$ = UJI BJI = -u $-x $I such that i%. =

at

BL

-

LB

the eigenvalue problem for L has an invariant spectrum and the Accordingly From eigenfunctions evolve in time according to a semigroup engendered by B. these remarks an approach motivated by an analogy with the inverse scattering