Gardner's deformation of the Krasil'shchik—Kersten system

Arthemy V Kiselev,Andrey O Krutov

doi:10.1088/1742-6596/621/1/012007

Abstract

The classical problem of construction of Gardner's deformations for infinite-dimensional completely integrable systems of evolutionary partial differential equations (PDE) amounts essentially to finding the recurrence relations between the integrals of motion. Using the correspondence between the zero-curvature representations and Gardner deformations for PDE, we construct a Gardner's deformation for the Krasil'shchik-Kersten system. For this, we introduce the new nonlocal variables in such a way that the rules to differentiate them are consistent by virtue of the equations at hand and second, the full system of Krasil'shchik-Kersten's equations and the new rules contains the Korteweg-de Vries equation and classical Gardner's deformation for it.

Highlights

We introduce the new nonlocal variables in such a way that the rules to differentiate them are consistent by virtue of the equations at hand and second, the full system of Krasil’shchik– Kersten’s equations and the new rules contains the Korteweg–de Vries equation and classical Gardner’s deformation for it
The search for conservation laws and, in particular, the search for regular methods of construction of conservation laws [1, 2] are the classical problems in the theory of infinitedimensional completely integrable systems
Mathieu’s N =2 SKdV super-equation was re-discovered; a recursion operator for symmetries of (1) was obtained via the introduction of suitable nonlocalities, c.f. [26] in this context. (Let us note that the coefficients of that recursion operator depend on the new nonlocal variables so that the locality of such operator’s output is arguable; yet it could well be that system (1) is but a precursor to the larger model with physical applications.) Around the same time, Karasu-Kalkanlı et al [27] approached system (1) with the Painleve test, performing the singularity analysis, and constructed an sl3(C)-valued zerocurvature representation α1KK for (1)

Summary

Introduction

The search for conservation laws and, in particular, the search for regular methods of construction of conservation laws [1, 2] are the classical problems in the theory of infinitedimensional completely integrable systems. (Let us note that the coefficients of that recursion operator depend on the new nonlocal variables so that the locality of such operator’s output is arguable; yet it could well be that system (1) is but a precursor to the larger model with physical applications.) Around the same time, Karasu-Kalkanlı et al [27] approached system (1) with the Painleve test, performing the singularity analysis, and constructed an sl3(C)-valued zerocurvature representation α1KK for (1) We shall use this Lie algebra-valued one-form for solving the Gardner’s deformation problem of recursive production of Hamiltonians for the hierarchy of the Krasil’shchik–Kersten system. We denote by Hn−1(E∞) the (n − 1)th horizontal cohomology group for E∞, that is, the set of equivalence classes of conserved currents which is equipped with the structure of Abelian group

Gardner’s deformations

The deformation of Krasil’shchik–Kersten’s system