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VOLUME 74 (2001) | ISSUE 10 | PAGE 541
On the initial-boundary value problems for soliton equations
PACS: 02.60.Lj, 03.40.Kf
We present a novel approach to solve initial-boundary value problems on the segment and on the half line for soliton equations. Our method is illustrated by solving a prototype, and widely applicable, 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, so obtaining a nonlinear integro - differential evolution equation for S(k,t). We also sketch an alternative approach, in the semiline case, based on 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.