A method for obtaining Darboux transformations

TL;DR

This paper introduces a systematic method for deriving Darboux transformations for integrable equations, demonstrated through examples including the dispersive water wave and Jaulent-Miodek equations.

Contribution

It presents a novel approach to obtain Darboux transformations, expanding the toolkit for analyzing integrable systems.

Findings

Derived Darboux transformation for dispersive water wave equation

Obtained Darboux transformation for Jaulent-Miodek equation via Miura map

Provides a general method applicable to various integrable equations

Abstract

In this paper we give a method to obtain Darboux transformations (DTs) of integrable equations. As an example we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation. \end{abstract}