The Robin and Neumann problems for the nonlinear Schr\"odinger equation on the half-plane

TL;DR

This paper establishes well-posedness for the 2D nonlinear Schrödinger equation on the half-plane with specific boundary conditions using explicit solution formulas and contraction mapping, advancing understanding of boundary value problems.

Contribution

It introduces a novel approach combining Fokas's unified transform with contraction mapping to prove well-posedness for nonlinear Schrödinger problems with Robin and Neumann boundary conditions.

Findings

Well-posedness in Sobolev spaces established

Explicit solution formulas derived via Fokas's method

Applicable to Robin and Neumann boundary conditions

Abstract

This work studies the initial-boundary value problem of the two-dimensional nonlinear Schr\"odinger equation on the half-plane with initial data in Sobolev spaces and Neumann or Robin boundary data in appropriate Bourgain spaces. It establishes well-posedness in the sense of Hadamard by utilizing the explicit solution formula for the forced linear initial-boundary value problem obtained via Fokas's unified transform, and a contraction mapping argument.