[Paper Review] Gardner's deformations of the N=2 supersymmetric a=4-KdV equation

V. Hussin, Alexander V. Kiselev+2 more|arXiv (Cornell University)|Nov 13, 2009

Nonlinear Waves and Solitons18 references5 citations

TL;DR

This paper proves the non-existence of supersymmetry-invariant Gardner's deformations for the $N=2$, $a=4$ supersymmetric KdV equation under standard retraction conditions. Instead, it proposes a two-step method: first deforming the bosonic Kaup–Boussinesq equation within the super-hierarchy, then using its recurrence structure to recursively generate the Hamiltonians of the full $N=2$, $a=4$–SKdV hierarchy, enabling systematic construction of integrals of motion for supersymmetric KdV-type systems.

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.

Motivation & Objective

• To resolve P. Mathieu's open problem on constructing supersymmetry-invariant Gardner's deformations for the $N=2$, $a=4$–KdV equation.
• To investigate whether known Gardner deformation techniques for scalar KdV can be extended to the $N=2$ supersymmetric case with $a=4$.
• To develop a systematic method for recursively generating integrals of motion in supersymmetric KdV-type hierarchies.
• To explore the role of bosonic limits and reductions in constructing deformations for the full super-hierarchy.
• To establish a generalizable framework applicable to other supersymmetric integrable systems beyond the $N=2$, $a=4$–SKdV equation.

Proposed method

• First, analyze the bosonic limit of the $N=2$, $a=4$–SKdV hierarchy, identifying the Kaup–Boussinesq equation as the second flow in the hierarchy.
• Construct a new polynomial, degree-$\leq 4$ Gardner’s deformation of the Kaup–Boussinesq system (25), with a cubic Miura contraction $\mathfrak{m}_{\epsilon}$.
• Derive the extended system $\mathcal{E}(\epsilon)$ (26b) that deforms the original Kaup–Boussinesq system while preserving the $1:1$ degree balance.
• Use the recurrence structure from the deformed Kaup–Boussinesq system to recursively generate the Hamiltonians $\boldsymbol{\mathcal{H}}^{(k)}$ of the full $N=2$, $a=4$–SKdV hierarchy.
• Apply the method to recover the super-Hamiltonians of the full super-hierarchy from the bosonic deformation, ensuring consistency with the original super-equation at $\epsilon=0$.
• Demonstrate that the method avoids the need for repeated application of the Leibniz rule by leveraging known residues and recurrence relations, making it more efficient than standard recursion operators.

Experimental results

Research questions

• RQ1 Does a supersymmetry-invariant Gardner’s deformation exist for the $N=2$, $a=4$–SKdV equation that retracts to the standard scalar KdV deformation under component reduction?
• RQ2 Can the recurrence of integrals of motion in the $N=2$, $a=4$–SKdV hierarchy be systematically constructed via a deformation of its bosonic limit?
• RQ3 Is the Kaup–Boussinesq equation a suitable intermediate system for generating the Hamiltonians of the full $N=2$, $a=4$–SKdV hierarchy through deformation?
• RQ4 What are the structural constraints on Miura-type contractions $\mathfrak{m}_{\epsilon}$ that prevent the existence of supersymmetry-invariant deformations in the full super-hierarchy?
• RQ5 Can the two-step deformation method be generalized to other supersymmetric KdV-type systems with similar bi-Hamiltonian structures?

Key findings

• There is no supersymmetry-invariant Gardner’s deformation for the $N=2$, $a=4$–SKdV equation that retracts to the standard scalar KdV deformation under component reduction.
• A unique polynomial Gardner’s deformation of degree $\leq 4$ was found for the two-component Kaup–Boussinesq system (25), with a cubic Miura contraction $\mathfrak{m}_{\epsilon}$ given by $u_{1} = \tilde{u}_{1}, \, u_{12} = \tilde{u}_{12} - \frac{1}{9}\epsilon^3 \tilde{u}_{1}\tilde{u}_{1;xx}$.
• The extended system $\mathcal{E}(\epsilon)$ (26b) is a consistent deformation of the Kaup–Boussinesq equation, with a correction term of order $\epsilon^3$ in the $u_{12;t}$ equation.
• The deformation of the Kaup–Boussinesq system yields a recurrence relation that allows the recursive construction of the super-Hamiltonians $\boldsymbol{\mathcal{H}}^{(k)}$ of the full $N=2$, $a=4$–SKdV hierarchy.
• The method bypasses the limitations of standard recursion operators by using known residues and recurrence relations, making it more efficient and systematic.
• The proposed two-step scheme is generalizable to other supersymmetric KdV-type systems, offering a robust framework for constructing integrals of motion in multi-Hamiltonian super-integrable systems.