Two-Component Coupled KdV Equations and its Connection with the Generalized Harry Dym Equation

TL;DR

This paper explores the connection between coupled KdV equations and the generalized Harry Dym equation through Lax operators, providing new integrable systems and Lax representations.

Contribution

It introduces a novel link between three generalized coupled Harry Dym equations and two-component KdV systems via reciprocal transformations.

Findings

Derived new Lax representations for coupled KdV and Harry Dym equations.

Connected three different Lax operators to known integrable systems.

Established reciprocal links transforming Harry Dym equations into coupled KdV systems.

Abstract

It is shown that, three different Lax operators in the Dym hierarchy, produce three generalized coupled Harry Dym equations. These equations transform, via the reciprocal link, to the coupled two-component KdV system. The first equation gives us known integrable two-component KdV system while the second reduces to the known symmetrical two-component KdV equation. The last one reduces to the Drienfeld-Sokolov equation. This approach gives us new Lax representation for these equations.