### Geometry and Dynamics Seminar

# Artemy Kiselev

### (Utrecht University)

## "Gardner's deformations of N=2 supersymmetric Korteweg-de Vries equations"

** Date: ** |
Tue, June 2, 2009 |

** Time: ** |
14:00 |

** Place: ** |
Research I Seminar Room |

**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 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.
However, we proceed with the study of the formal wave function and the tau
function for 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.
(This is a joint work with V. Hussin, Montreal.)