Symmetries for exact solutions to the nonlinear Schr\"odinger equation

TL;DR

This paper presents a method to derive explicit soliton solutions for the nonlinear Schr"odinger equation using symmetry and matrix triplets, applicable to scalar and matrix cases, and relates these solutions to spectral data.

Contribution

It introduces a symmetry-based approach to express exact solutions of the nonlinear Schr"odinger equation in terms of constant matrix triplets, generalizing to matrix equations and linking solutions to spectral properties.

Findings

Explicit soliton solutions in terms of matrix triplets

Derivation of spectral data from soliton solutions

Extension to matrix nonlinear Schr"odinger equations

Abstract

A certain symmetry is exploited in expressing exact solutions to the focusing nonlinear Schr\"odinger equation in terms of a triplet of constant matrices. Consequently, for any number of bound states with any number of multiplicities the corresponding soliton solutions are explicitly written in a compact form in terms of a matrix triplet. Conversely, from such a soliton solution the corresponding transmission coefficients, bound-state poles, bound-state norming constants and Jost solutions for the associated Zakharov-Shabat system are evaluated explicitly. It is also shown that these results hold for the matrix nonlinear Schr\"odinger equation of any matrix size.