On the initial-boundary value problems for soliton equations

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.