The boundary value problem for the nonlinear Schrödinger equation

A boundary value problem on the half-line for the nonlinear Schrödinger equation (NLS) and its generalisation is studied by the inverse scattering transform method. The connection between conservation laws and boundary conditions for integrable boundary value problems for higher NLS equations is established. It's shown that the generalized boundary value problem requires a joint analysis of regular and singular solutions for the NLS equations in the repulsive case.