Integrability and rational soliton solutions for gauge invariant derivative nonlinear Schr\"odinger equations

TL;DR

This paper investigates the integrability of three derivative nonlinear Schrödinger equations, derives their Lax pairs, and constructs rational soliton solutions using Darboux transformations, advancing understanding of their mathematical structure and solution space.

Contribution

It introduces new Lax pairs for these equations and develops a method to generate rational soliton solutions, enhancing the analytical tools for derivative nonlinear Schrödinger equations.

Findings

Successfully obtained Lax pairs via Miura transformation and singular manifold method.

Constructed rational soliton-like solutions for the equations.

Provided a framework for analyzing integrability and solutions of derivative NLS equations.

Abstract

The present work addresses the study and characterization of the integrability of three famous nonlinear Schr\"odinger equations with derivative-type nonlinearities in 1+1 dimensions. Lax pairs for these three equations are successfully obtained by means of a Miura transformation and the singular manifold method. After implementing the associated binary Darboux transformations, we are able to construct rational soliton-like solutions for those systems.