Robin boundary condition and shock problem for the focusing nonlinear Schr\"odinger equation

Abstract

We consider the initial boundary value (IBV) problem for the focusing nonlinear Schr\"odinger equation in the quarter plane $x>0,t>0$ in the case of periodic initial data (at $t=0$) and a Robin boundary condition at $x=0$. Our approach is based on the simultaneous spectral analysis of the Lax pair equations combined with symmetry considerations for the corresponding Riemann-Hilbert problems. A connection between the original IBV problem and an associated initial value (IV) problem is established.