Colloquium Mathematics, Drs. A.O. Krutov (RuG)

10 June 2014

Join us for coffee and tea at 15.45 p.m.

Date:	Tuesday, June 10th 2014
Speaker:	Drs. A.O. Krutov (RuG)
Room:	5161.0293 (Bernoulliborg)
Time:	16.00

Title: Gardner's deformations and zero-curvature representations for KdV-like systems.

Abstract:

The deformation approach is an important tool in the study of partial differential equations (PDE). Gardner’s deformations are an example of such concept. By definition, Gardner’s deformation is a family of pairs consisting of deformation of equation and Miura’s map which takes solutions of deformed equation to solutions of the original equation. Using Gardner’s deformation, one can recover a recurrence relation between the Hamiltonians of a given PDE. We consider the problem of construction of Gardner's deformation for various generalisations of the Korteweg-de Vries (KdV) equation: namely, the Krasil'shchik-Kersten system (which is a coupled system of KdV and mKdV equations) and the N=2 supersymmetric a=4 Korteweg-de Vries equation (which is a generalisation of the KdV equation with unknown function depending also on anticommuting Grassmann variables $\theta_1$, $\theta_2$). Zero-curvature representations for PDE are another construction in the geometry of PDE. By definition, a zero-curvature representation is a Lie algebra-valued horizontal differential one-form satisfying the Maurer-Cartan condition. Lie algebra-valued zero-curvature representations for PDE are the input data for solving Cauchy’s problems by the inverse scattering method. Gardner's deformations and zero-curvature representations can be considered in the unifying framework. Indeed, both constructions yield systems of new nonlocal variables such that their derivatives commute by virtue of the original equation. Using this relations between zero-curvature representations and Gardner's deformations, we solve the Gardner deformation problems for the N=2 supersymmetric a=4 Korteweg-deVries equation and Krasil'shchik-Kersten system.

Colloquium coordinators are Prof.dr. A.C.D. van Enter (e-mail : A.C.D.van.Enter@rug.nl) and
Dr. A.V. Kiselev (e-mail: a.v.kiselev rug.nl )

http://www.rug.nl/research/jbi/news/colloquia/mathematics-colloquia/

Last modified:10 February 2021 2.28 p.m.

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