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

TL;DR

This paper constructs a Gardner's deformation for the Krasil'shchik-Kersten system by introducing new nonlocal variables, linking it to the classical Gardner deformation for the Korteweg-de Vries equation, and advancing the understanding of integrable PDEs.

Contribution

It introduces a novel Gardner's deformation for the Krasil'shchik-Kersten system using zero-curvature representations and nonlocal variables, connecting it to classical integrable systems.

Findings

Constructed Gardner's deformation for the Krasil'shchik-Kersten system.

Linked the deformation to the classical Korteweg-de Vries equation.

Established recurrence relations between integrals of motion.

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. PACS: 02.30.Ik, 02.30,Jr, 02.40.-k, 11.30.-j