Enhanced group classification of Gardner equations with time-dependent coefficients

TL;DR

This paper provides an exhaustive classification of Lie symmetries for variable coefficient Gardner equations, enhancing understanding of their symmetry properties and deriving classifications for related mKdV equations with forcing.

Contribution

It introduces a comprehensive group classification method for variable coefficient Gardner equations using equivalence transformations and mapping techniques, extending previous partial results.

Findings

Complete group classification achieved using equivalence transformations.

Derived classification for variable coefficient mKdV equations with forcing.

Discussed advantages of generalized extended equivalence groups.

Abstract

We classify the Lie symmetries of variable coefficient Gardner equations (called also the combined KdV-mKdV equations). In contrast to the particular results presented in Molati and Ramollo (2012) we perform the exhaustive group classification. It is shown that the complete result can be achieved using either the gauging of arbitrary elements of class by the equivalence transformations or the method of mapping between classes. As by-product of the second approach the complete group classification of a class of variable coefficient mKdV equations with forcing term is derived. Advantages of the use of the generalized extended equivalence group in comparison with the usual one are also discussed.