Initial boundary value problems for integrable systems: towards the long time asymptotics

The long time behaviour of solutions of (Dirichlet) initial boundary value problems for the focusing nonlinear Schrödinger equation in the case of (asymptotically) periodic boundary conditions of special (one-frequency) structure is considered both theoretically and numerically. The results of numerical simulations are shown to confirm the theoretical description in the parameter ranges where the assumption about the one-frequency character of the Neumann boundary values is (theoretically) admissible, whereas in the other ranges they suggest a more complicated description of the behaviour of these values