Initial boundary value problem for the focusing NLS equation with Robin boundary condition: half-line approach

TL;DR

This paper addresses the initial boundary value problem for the focusing nonlinear Schrödinger equation on a half-line with Robin boundary conditions, utilizing spectral analysis without assuming long-term boundary behavior.

Contribution

It demonstrates that the spectral approach can be applied without prior assumptions on boundary value long-time behavior for the focusing NLS with Robin boundary conditions.

Findings

Spectral analysis method applied without a priori boundary assumptions

Effective handling of focusing NLS on a half-line with Robin boundary

Extension of Fokas' approach to new boundary conditions

Abstract

We consider the initial boundary value problem for the focusing nonlinear Schr\"odinger equation in the quarter plane $x>0,t>0$ in the case of decaying initial data (for $t=0$, as $x\to +\infty$) and the Robin boundary condition at $x=0$. We revisit the approach based on the simultaneous spectral analysis of the Lax pair equations [Fokas, 2002; Fokas, Its, Sung, 2005] and show that the approach can be implemented without any a priori assumptions on the long-time behavior of the boundary values.