Derivative non-linear Schr\"odinger equation: Singular manifold method and Lie symmetries

TL;DR

This paper investigates the integrability of the derivative nonlinear Schrödinger equation using the singular manifold method, Lie symmetries, and Darboux transformations to derive solutions and analyze its properties.

Contribution

It introduces a generalized approach to study the integrability of the derivative NLS equation, deriving a Lax pair and constructing rational soliton solutions.

Findings

Derived a Lax pair via Miura transformation and singular manifold method.

Constructed a broad class of rational soliton-like solutions.

Analyzed Lie symmetries and performed similarity reductions.

Abstract

We present a generalized study and characterization of the integrability properties of the derivative non-linear Schr\"odinger equation in 1+1 dimensions. A Lax pair is derived for this equation by means of a Miura transformation and the singular manifold method. This procedure, together with the Darboux transformations, allow us to construct a wide class of rational soliton-like solutions. Lie classical symmetries have also been computed and similarity reductions have been analyzed and discussed.