Enhanced group classification of Gardner equations with time-dependent coefficients

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 results can be achieved using either the gauging of arbitrary elements of the 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. © 2014 Elsevier B.V.