Integrable and nonintegrable initial boundary value problems for soliton equations

It is well-known that the basic difficulty in studying the initial boundary value problems for linear and nonlinear PDEs is the presence, in any method of solution, of unknown boundary values. In the first part of this paper we review two spectral methods in which the above difficulty is faced in different ways. In the first method one uses the analyticity properties of the x-scattering matrix S(k, t) to replace the unknown boundary values by elements of the scattering matrix itself, thus obtaining a closed integro-differential evolution equation for S(k, t). In the second method one uses the analyticity properties of S(k, t) to eliminate the unknown boundary values by a suitable projection, obtaining a nonlinear Riemann Hilbert problem for S(k, t). The second approach allows also to identify in a natural way a known subclass of boundary conditions which gives rise to a spectral formalism based on linear operations (and therefore called "integrable boundary conditions" ). In the last part of the paper we present a new method to identify a whole hierarchy of integrable boundary conditions.