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

Group classification of classes of mKdV-like equations with time-dependent coefficients is carried out. The usage of equivalence transformations appears to be 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 for the construction of exact solutions of mKdV-like equations using equivalence transformations is described.

► Group classifications of two variable coefficient mKdV equations are carried out.
► Usage of equivalence is the crucial point in exhaustive solving the problem.
► Exact solutions for equations reducible to standard mKdV equation are constructed.