Initial-boundary value problems for the defocusing nonlinear Schr\"odinger equation in the semiclassical limit

TL;DR

This paper develops an explicit approximation method for solving initial-boundary value problems of the defocusing nonlinear Schrödinger equation, bypassing the global relation and providing accurate semiclassical limit solutions for nonhomogeneous boundary conditions.

Contribution

It introduces a novel approximation for the nonlinear Dirichlet-to-Neumann map, enabling solutions without solving the global relation for general boundary conditions.

Findings

Method provides accurate semiclassical asymptotics.

Circumvents the need to solve the global relation.

Applicable to nonhomogeneous boundary conditions.

Abstract

Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required make the problem well-posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so-called global relation, and types of boundary conditions for which the global relation can be solved are called \emph{linearizable}. For the defocusing nonlinear Schr\"odinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schr\"odinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial-boundary value problem on the half-line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter-plane space-time domain.