A non-standard Lax formulation of the Harry Dym hierarchy and its supersymmetric extension

TL;DR

This paper introduces a novel non-standard Lax formulation for the Harry Dym hierarchy derived from KdV, and constructs a new N=2 supersymmetric extension linked to supersymmetric KdV equations, enriching the integrable systems framework.

Contribution

It presents a new non-standard Lax formulation for the Harry Dym hierarchy and develops an N=2 supersymmetric extension connected to supersymmetric KdV equations.

Findings

Derived a non-standard Lax formulation from KdV via reciprocal transformation.

Constructed a new N=2 supersymmetric Harry Dym hierarchy.

Linked the supersymmetric Harry Dym equation to KdV-MKdV systems.

Abstract

For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of Korteweg-de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N=2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be linked to one of the N=2 supersymmetric KdV equations through superconformal transformation. The bosonic limit of this new N=2 supersymmetric Harry Dym equation is related to a coupled system of KdV-MKdV equations.