Algebraic Properties of Gardner’s Deformations for Integrable Systems

An algebraic definition of Gardner’s deformations for completely integrable bi-Hamiltonian evolutionary systems is formulated. The proposed approach extends the class of deformable equations and yields new integrable evolutionary and hyperbolic Liouville-type systems. An exactly solvable two-component extension of the Liouville equation is found. Introduction. We consider the problem of constructing Gardner’s deformations of completely integrable bi-Hamiltonian evolutionary (super-)systems, see [1, 2, 3] or the review [4] and references therein. The deformations yield recurrence relations for densities of the Hamiltonians and result in parametric extensions of known equations. Various methods of constructing these deformations were addressed in classical and supersymmetric settings from various standpoints [3, 4, 5], and yet no comprehensive idea allows to explain where the deformations arise from, how they can be found, and what the obstructions for their existence are. One of the difficulties of the theory was implied by the emphasis on constructing families of integrable equations, although the principal idea of Gardner’s deformations is that they specify the Hamiltonians, thus proving the complete integrability and the locality of hierarchies. Also, the well-known deformations of KdV equations [1, 6] created a belief that the integrable extensions interpolate towards the modified systems [5] and that the Galilean invariance plays the central role [4]. The objective of this work is the detailed analysis of obstructions for existence of Gardner’s deformations of integrable (super-)systems; in particular, we bear in mind the open problems of deforming the N = 2 Super-KdV equations [2, 5]. In this paper we propose an algebraic definition of Gardner’s deformations; it simplifies their search. Within this approach, the Miura-type transformations that act from a system under study are found first. Thus we profit from the relation between the integrals of Liouville-type equations [7] and non-invertible differential substitutions; this continues the line of reasonings in [8]. Considering the KdV and Kaup-Boussinesq equations, we show that Miura’s transformations from the modified hierarchies determine new exactly solvable Liousillve-type systems and that parametric extensions of Date: October 30, 2006. 2000 Mathematics Subject Classification. 35Q53, 37K05, 37K10, 37K35.