Gardner's deformations of N = 2 supersymmetric Korteweg–de Vries equations

Abstract:
We consider the problem of constructions of continuous integrable deformations for P. Mathieu's N = 2 supersymmetric KdV equations. These evolutionary systems incorporate the standard KdV equation, but now the unknown function takes values in the four-dimensional Grassmann algebra. The purpose of constructing such deformations is two-fold: they yield recurrence relations between the Hamiltonians of the corresponding hierarchy and, on the other hand, specify the Lax representations which are used further for solving the Cauchy problem by the inverse scattering. Analyzing the equivalence of the deformations and the Lax representations, we extend the ''no-go'' result on Gardner's deformations in the classical sense for N = 2 SKdV to the "no-know" claim about the Lax pair for it. This is due to the supersymmetric setup and does not have an analogue in the commutative theory. Still we obtain sufficiently many deformations for the bosonic limit of the N = 2 SKdV. However, we proceed with the study of the formal wave function and the tau function for the N = 2 SKdV and find a new class of its exact solutions. These Hirota's supersolitons possess paradoxal properties. Namely, they do not accumulate any phase shifts during the elastic scattering, and they can be subject to a spontaneous decay followed by the transition into virtual states.

References:
[1] A.V. Kiselev (2007) Algebraic properties of Gardner's deformations for integrable systems, Theoret. Math. Phys. 152:(1), 963--976.
[2] A.V. Kiselev, V. Hussin (2009) Hirota's virtual multisoliton solutions of N = 2 supersymmetric Korteweg–de Vries equations, Theoret. Math. Phys. 159:(3), 832--840.