Gardner's deformations of the graded Korteweg-de Vries equations revisited

TL;DR

This paper revisits Gardner's deformations for the graded Korteweg-de Vries equations, specifically solving the deformation problem for the N=2 supersymmetric a=4 KdV equation and linking it to classical KdV via zero-curvature representations.

Contribution

It provides a solution to the Gardner deformation problem for the N=2 supersymmetric a=4 KdV equation and connects it to classical KdV through zero-curvature representations.

Findings

Derived new nonlocal variables for the supersymmetric KdV

Linked supersymmetric Gardner deformation to classical KdV

Extended understanding of integrability structures in supersymmetric equations

Abstract

We solve the Gardner deformation problem for the N=2 supersymmetric a=4 Korteweg-de Vries equation (P. Mathieu, 1988). We show that a known zero-curvature representation for this superequation yields the system of new nonlocal variables such that their derivatives contain the Gardner deformation for the classical KdV equation.