Yang-Baxter and reflection maps from vector solitons with a boundary

TL;DR

This paper develops a mathematical framework for understanding how vector solitons interact with boundaries, introducing reflection maps and solving related equations, advancing the theory of integrable boundary conditions in nonlinear wave equations.

Contribution

The authors introduce reflection maps satisfying set-theoretical reflection equations, providing new solutions and insights into soliton-boundary interactions for vector nonlinear Schrödinger equations.

Findings

Factorization of soliton interactions on the half-line is established.

New reflection maps satisfying reflection equations are constructed.

Theoretical foundations of set-theoretical reflection equations are developed.

Abstract

Based on recent results obtained by the authors on the inverse scattering method of the vector nonlinear Schr\"odinger equation with integrable boundary conditions, we discuss the factorization of the interactions of N-soliton solutions on the half-line. Using dressing transformations combined with a mirror image technique, factorization of soliton-soliton and soliton-boundary interactions is proved. We discover a new object, which we call reflection map, that satisfies a set-theoretical reflection equation which we also introduce. Two classes of solutions for the reflection map are constructed. Finally, basic aspects of the theory of set-theoretical reflection equations are introduced.