Well-Posedness of the Nonlinear Schr\"odinger Equation on the Half-Plane

TL;DR

This paper establishes local well-posedness for the nonlinear Schrödinger equation on the half-plane with nonzero boundary data, using a novel approach based on the Fokas unified transform and Sobolev/Bourgain spaces.

Contribution

It introduces a new method for analyzing the IBVP of NLS on the half-plane, extending well-posedness results to higher dimensions with nonzero boundary conditions.

Findings

Proves local well-posedness for NLS on the half-plane.

Utilizes the Fokas unified transform for solution representation.

Handles boundary data in Bourgain spaces.

Abstract

The initial-boundary value problem (IBVP) for the nonlinear Schr\"odinger (NLS) equation on the half-plane with nonzero boundary data is studied by advancing a novel approach recently developed for the well-posedness of the cubic NLS on the half-line, which takes advantage of the solution formula produced by the unified transform of Fokas for the associated linear IBVP. For initial data in Sobolev spaces on the half-plane and boundary data in Bourgain spaces arising naturally when the linear IBVP is solved with zero initial data, the present work provides a local well-posedness result for NLS initial-boundary value problems in higher dimensions.