Seminar talk, 16 September 2009

Speaker: Arthemy Kiselev

Title: A new class of exact solutions and integrable deformations of $N=2$ supersymmetric KdV-Mathieu equation

Abstract:

The talk based on a joint work with Veronique Hussin (see a reference below) and on yet unpublished joint work with Veronique Hussin and Tomas Wolf.

There will be constructed a new class of exact multi-soliton solutions in Hirota's form of Mathieu's supersymmetric Korteweg-de Vries equations with parameter $a=1$ (complicated case) and $a=4$ (superintegrable case). The interaction of solitons is nontrivial: unlike solutions inherited from KdV and subject to the usual superposition laws with phase shifts, in our case there are no phase shifts but, for an observer, the solitons spontaneously decay (become virtual) or are created (from exponentially small perturbation of visible ones).

The above paradoxal properties "respect" the infinite set of conservation laws of the corresponding integrable hierarchy. To obtain Hamiltonians in a recurrent way for the superintegrable case $a=4$, we construct an integrable deformation by applying three nontrivial auxiliary constructions in succession, but without explicit use of the equivalence of the problem of deformation and zero curvature representation. Thus, in the talk there will be presented a solution of a problem posed by Mathieu in 1989.

References:

Arthemy V. Kiselev and Veronique Hussin, Hirota's virtual multi-soliton solutions of $N=2$ supersymmetric Korteweg-de Vries equations, Theor. Math. Phys. 159 (2009), 833-841, arXiv:0810.0930; Russian original: Teor. Mat. Fiz. 159 (2009), 490-501, Mi tmf6367