Darboux transformation of the generalized coupled dispersionless integrable system

TL;DR

This paper investigates the Darboux transformation for a generalized coupled dispersionless integrable system, expressing solutions via quasideterminants and determinants, with explicit examples for the SU(2) Lie group.

Contribution

It introduces a method to construct solutions of the system using Darboux transformations and quasideterminants, extending previous approaches to non-abelian Lie groups.

Findings

Solutions expressed in quasideterminants for non-abelian cases

Explicit solutions for SU(2) case as ratios of determinants

Demonstrates the Darboux transformation's applicability to generalized systems

Abstract

The Darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on some non-abelian Lie group, is studied and the solutions are shown to be expressed in terms of quasideterminants. As an explicit example, the Darboux transformation on scalar solutions to the system based on the Lie group SU(2) is discussed in detail and the solutions are shown to be expressed as ratios of determinants.