Two-component coupled KdV equations and its connection with the generalized Harry Dym equations

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 Korteweg de Vries (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.