On the nonlinear Schr\"odinger equation with a time-dependent boundary condition

TL;DR

This paper investigates the nonlinear Schrödinger equation on a half-line with a novel time-dependent boundary condition, establishing its integrability and deriving explicit multi-soliton solutions using advanced mathematical techniques.

Contribution

It introduces a new integrable boundary condition involving time derivatives and develops methods to construct explicit multi-soliton solutions for this boundary problem.

Findings

Proves integrability of the boundary condition using Sklyanin's formalism.

Constructs explicit multi-soliton solutions via Darboux transformations.

Provides a systematic approach for boundary dressing techniques.

Abstract

We study the nonlinear Schr\"odinger equation on the half-line with a boundary condition that involves time derivative. This boundary condition was presented by Zambon [J. High Energ. Phys. 2014 (2014) 36]. We establish the integrability of such a boundary both by using the Sklyanin's formalism and by using the tool of B\"acklund transformations together with a suitable reduction of reflection type. Moreover, we present a method to derive explicit formulae for multi-soliton solutions of the boundary problem by virtue of the Darboux transformation method in conjunction with a boundary dressing technique.