Approximate Hamiltonian Symmetry Groups and Recursion Operators for Perturbed Evolution Equations

TL;DR

This paper extends the method of approximate transformation groups to Hamiltonian systems, enabling the computation of approximate conservation laws and recursion operators, with applications to the Gardner equation.

Contribution

It introduces an extended procedure for approximate symmetry analysis specifically for Hamiltonian and bi-Hamiltonian evolution equations.

Findings

Derived approximate conservation laws for Hamiltonian systems.

Computed approximate recursion operators for perturbed equations.

Applied methods to the Gardner equation with small parameters.

Abstract

In this paper, the method of approximate transformation groups which was proposed by Baikov, Gazizov and Ibragimov, is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and approximate recursion operators corresponding to these types of equations. In particular, as an application, a comprehensive analysis of the problem of approximate conservation laws and approximate recursion operators associated to the Gardner equation with the small parameters is presented.