Geometric study of Gardner equation

TL;DR

This paper applies approximate symmetry methods to the Gardner equation, deriving invariant solutions and analyzing symmetries, including approximate Galilean invariance, to deepen understanding of its geometric properties.

Contribution

It introduces a systematic approach to find approximate symmetries and invariant solutions for the Gardner equation using the method of approximate transformation groups.

Findings

Computed first-order approximate symmetries of the Gardner equation.

Derived the optimal system of symmetries.

Obtained general forms of approximately Galilean-invariant solutions.

Abstract

In this paper, we apply the method of approximate transformation groups proposed by Baikov, Gaziziv and Ibragimov, to compute the first-order approximate symmetry for the Gardner equations with the small parameters. We compute the optimal system and analyze some invariant solutions of These types of equations. Particularly, general forms of approximately Galilean-invariant solutions have been computed.