Gardner's deformations of the N=2 supersymmetric a=4-KdV equation

TL;DR

This paper investigates the existence of Gardner's deformation for the N=2 supersymmetric a=4-KdV equation, proving non-existence under certain conditions, and introduces a new deformation for the related Kaup-Boussinesq equation to generate integrals of motion.

Contribution

It proves the non-existence of supersymmetry-invariant Gardner's deformations for the N=2 a=4-KdV under specific assumptions and proposes a new recursive scheme using a deformation of the Kaup-Boussinesq equation.

Findings

No supersymmetry-invariant solutions exist for the Gardner's deformation of N=2 a=4-KdV under component reduction.

A new Gardner's deformation for the Kaup-Boussinesq equation is constructed.

The method enables recursive generation of integrals of motion for the N=2, a=4-SKdV hierarchy.

Abstract

We prove that P.Mathieu's Open problem on constructing Gardner's deformation for the N=2 supersymmetric a=4-Korteweg-de Vries equation has no supersymmetry invariant solutions, whenever it is assumed that they retract to Gardner's deformation of the scalar KdV equation under the component reduction. At the same time, we propose a two-step scheme for the recursive production of the integrals of motion for the N=2, a=4-SKdV. First, we find a new Gardner's deformation of the Kaup-Boussinesq equation, which is contained in the bosonic limit of the super-hierarchy. This yields the recurrence relation between the Hamiltonians of the limit, whence we determine the bosonic super-Hamiltonians of the full N=2, a=4-SKdV hierarchy. Our method is applicable towards the solution of Gardner's deformation problems for other supersymmetric KdV-type systems.