Soliton solutions of the nonlinear Schr\"odinger equation with defect conditions

TL;DR

This paper develops a method to construct and analyze soliton solutions of the nonlinear Schrödinger equation with defect conditions, showing solitons transmit through defects independently, advancing understanding of integrable systems with boundaries.

Contribution

It introduces a Darboux transformation approach to generate N-soliton solutions with defect conditions, preserving spectral constraints and demonstrating independent soliton transmission.

Findings

Constructed N-soliton solutions with defect conditions.

Proved solitons transmit independently through the defect.

Preserved spectral boundary constraints with a time-dependent defect matrix.

Abstract

A recent development in the derivation of soliton solutions for initial-boundary value problems through Darboux transformations, motivated to reconsider solutions to the nonlinear Schr\"odinger (NLS) equation on two half-lines connected via integrable defect conditions. Thereby, the Darboux transformation to construct soliton solutions is applied, while preserving the spectral boundary constraint with a time-dependent defect matrix. In this particular model, $N$-soliton solutions vanishing at infinity are constructed. Further, it is proven that solitons are transmitted through the defect independently of one another.