On the initial-boundary value problems for soliton equations

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Abstract

We present a novel approach to solving initial-boundary value problems on the segment and the half line for soliton equations. Our method is illustrated by solving a prototypal and widely applied dispersive soliton equation—the celebrated nonlinear Schroedinger equation. It is well known that the basic difficulty associated with boundaries is that some coefficients of the evolution equation of the (x) scattering matrix S(k, t) depend on unknown boundary data. In this paper, we overcome this difficulty by expressing the unknown boundary data in terms of elements of the scattering matrix itself to obtain a nonlinear integrodifferential evolution equation for S(k, t). We also sketch an alternative approach in the semiline case on the basis of a nonlinear equation for S(k, t), which does not contain unknown boundary data; in this way, the “linearizable” boundary value problems correspond to the cases in which S(k, t) can be found by solving a linear Riemann-Hilbert problem.

From Pis’ma v Zhurnal Éksperimental’no \(\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{l} \) i Teoretichesko \(\overset{\lower0.5em\hbox{\(\smash{\scriptscriptstyle\smile}\)}}{l} \) Fiziki, Vol. 74, No. 10, 2001, pp. 541–545.
Original English Text Copyright © 2001 by Degasperis, Manakov, Santini.
This article was submitted by the authors in English.