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

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.

Extended version of the talks given by A.V.K. at 8th International conference `Symmetry in Nonlinear Mathematical Physics' (June 20-27, 2009, Kiev, Ukraine) and 9th International workshop `Supersymmetry and Quantum Symmetries' (July 29 - August 3, 2009, JINR, Dubna, Russia); 22 pages

References in corpus (4)

• A simple way of making a Hamiltonian system into a bi-Hamiltonian one
• Classification of integrable super-systems using the SsTools environment
• Algebraic properties of Gardner's deformations for integrable systems
• Variational Poisson-Nijenhuis structures for partial differential equations

Cited by in corpus (7)

• On the non-abelian superalgebra spanned by the conserved quantities of N=1 supersymmetric Korteweg-de Vries equation
• Gardner's deformations of the graded Korteweg-de Vries equations revisited
• Gardner's deformation of the Krasil'shchik-Kersten system
• On the (non)removability of spectral parameters in $Z_2$-graded zero-curvature representations and its applications
• Gardner's deformations as generators of new integrable systems
• A novel Hirota bilinear approach to $N=2$ supersymmetric equations
• A convenient criterion under which Z_2-graded operators are Hamiltonian