Symmetry in Nonlinear Mathematical Physics - 2009
(Université de Montréal, Canada)
Arthemy Kiselev (Ivanovo State Power University, Russia & Utrecht University, The Netherlands)
Gardner's deformations of N = 2 supersymmetric Korteweg–de Vries equations
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.
 A.V. Kiselev (2007) Algebraic properties of Gardner's deformations
for integrable systems, Theoret. Math. Phys. 152:(1), 963--976.
 A.V. Kiselev, V. Hussin (2009) Hirota's virtual multisoliton solutions
of N = 2 supersymmetric Korteweg–de Vries equations, Theoret. Math. Phys.