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

TL;DR

This paper analyzes the initial boundary value problem for the focusing nonlinear Schrödinger equation with Robin boundary conditions, using spectral analysis and Riemann-Hilbert techniques to connect it with an associated initial value problem.

Contribution

It introduces a spectral approach for the IBV problem with Robin boundary conditions, linking it to an IV problem via symmetry considerations.

Findings

Spectral analysis of Lax pair equations for the IBV problem.

Establishment of a connection between IBV and IV problems.

Application of Riemann-Hilbert problem techniques.

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.