Initial boundary value problem for the focusing nonlinear Schr?dinger equation with Robin boundary condition: Half-line approach

摘要

We consider the initial boundary value problem for the focusing nonlinear Schr?dinger equation in the quarter plane x < 0, t < 0 in the case of decaying initial data (for t =0, as x →+∞) and the Robin boundary condition at x=0. We revisit the approach based on the simultaneous spectral analysis of the Lax pair equations and show that the method can be implemented without any a priori assumptions on the long-time behaviour of the boundary values.