An integrable generalization of the nonlinear Schr\"odinger equation on the half-line and solitons

TL;DR

This paper studies an integrable extension of the nonlinear Schrödinger equation on a half-line, focusing on boundary conditions and explicit solution verification to advance understanding of soliton behavior.

Contribution

It introduces and analyzes a new integrable generalization of the nonlinear Schrödinger equation with Robin boundary conditions on the half-line.

Findings

Explicit solution verification demonstrates the well-posedness of the problem.

Analysis of linearizable boundary conditions for the integrable system.

Clarification of soliton dynamics under new boundary conditions.

Abstract

We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schr\"odinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem.