Lie symmetries and exact solutions of variable coefficient mKdV equations: an equivalence based approach

TL;DR

This paper classifies variable coefficient mKdV equations using equivalence transformations, providing a systematic approach to find exact solutions and demonstrating the normalization of these classes for comprehensive analysis.

Contribution

It introduces an equivalence-based framework for classifying and solving variable coefficient mKdV equations, highlighting the normalization property of these classes.

Findings

All classes are normalized, facilitating classification.

Exact solutions can be constructed via equivalence transformations.

Classification results are provided up to different equivalence types.

Abstract

Group classification of classes of mKdV-like equations with time-dependent coefficients is carried out. The usage of equivalence transformations appears a crucial point for the exhaustive solution of the problem. We prove that all the classes under consideration are normalized. This allows us to formulate the classification results in three ways: up to two kinds of equivalence (which are generated by transformations from the corresponding equivalence groups and all admissible point transformations) and using no equivalence. A simple way of the construction of exact solutions of mKdV-like equations using equivalence transformations is described.