N = 2 supersymmetric a=4-Korteweg–de Vries hierarchy derived via Gardner’s deformation of Kaup–Boussinesq equation

Abstract

We consider the problem of constructing Gardner's deformations for the N =2 supersymmetric a = 4-Korteweg-de Vries ͑SKdV͒ equation; such deformations yield recurrence relations between the super-Hamiltonians of the hierarchy. We prove the nonexistence of supersymmetry-invariant deformations that retract to Gardner's formulas for the Korteweg-de Vries ͑KdV͒ with 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 superhierarchy. 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 toward the solution of Gardner's deformation problems for other supersymmetric KdV-type systems.

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

“…By now the Gardner deformation problem for the N=2 supersymmetric a=4 Korteweg-de Vries equation ( [3]) is solved. In this paper we have found the solution which is an alternative to our previous result in [4]. Namely, we introduced the nonlocal bosonic and fermionic variables in such a way that the rules to differentiate them are consistent by virtue of the super-equation at hand and second, the entire system retracts to the standard KdV equation and the classical Gardner deformation for it ( [1]) under setting to zero the fermionic nonlocal variable and the first three components of the N=2 superfield in (29).…”

“…In section 2 we proceed with this correspondence for Z 2 -graded systems of evolutionary PDE and solve the Gardner deformation problem for the N=2, a=4 SKdV (29). The nature of the new solution is geometric and it presents an alternative to the analytic two-step algorithm that works for graded systems and which we reported earlier in [4].…”

“…Let us consider the bosonic limit u 1 = u 2 = 0 of system (31): by setting a = −2 we obtain the triangular system which consists of the modified KdV equation upon u 0 and the equation of KdV-type; in the case a = 1 we obtain the Krasil'shchik-Kersten system; for a = 4, we obtain the third equation in the Kaup-Boussinesq hierarchy. A Gardner deformation of the Kaup-Boussinesq system was constructed in [4]. The Gardner deformation problem for the N = 2 supersymmetric a = 4 KdV equation was formulated by P. Mathieu in [2].…”

“…The Gardner deformation problem for the N = 2 supersymmetric a = 4 KdV equation was formulated by P. Mathieu in [2]. In the paper [4] it was shown that one can not construct such a deformation under the assumptions that, first, the deformation is polynomial in E, second, it involves only the super-fields but not their components, and third, it contains the known deformation (11) under the reduction u 0 = 0, u 1 = u 2 = 0. Therefore, we shall find a graded generalization of Gardner's deformation (11) for the system of four equations (31) treating it in components but not as the single equation (29) upon the super-field.…”

“…Various attempts to solve it were undertaken since then (e.g., see [2]) but the progress was limited. The first solution for the N=2, a=4 SKdV in the triplet a ∈ {−2, 1, 4} was achieved in [4]; in that paper, we stated the 'no-go' theorem about the impossibility to deform this super-equation in terms of the superfield (but not the impossibility to deform it by treating the components of the super-field separately, see below), c.f. [2].…”