On A New Form of Darboux-B\"acklund Transformation for DNLS Equation-Mixed and Rational Type Solutions

TL;DR

This paper introduces a novel Darboux-Bäcklund transformation for the DNLS equation, enabling the generation of mixed and rational solutions through different methodologies, expanding the solution space of the equation.

Contribution

A new form of Darboux-Bäcklund transformation for DNLS is derived, allowing for the construction of mixed and rational solutions using distinct approaches.

Findings

A new Darboux-Bäcklund transformation form for DNLS.

Generation of mixed solutions with algebraic and exponential dependence.

Construction of purely rational solutions using Neugebauer's methodology.

Abstract

A new form of Darboux-B\"acklund transformation and its higher order form is derived for Derivative Nonlinear Schrodinger Equation(DNLS). The new form arises due to the different form of Lax pair. It is observed that by a special choice of the eigenvalue of DB transformation one can generate a mixed form of solution(containing both algebraic and exponential dependence on (x, t) can be generated. On the other hand by adopting a new methodology due to Neugebauer et. al. it is found that purely rational solution can be constructed. The two different approach yields different class of solution and are compared.