Group Analysis of Variable Coefficient Diffusion-Convection Equations. I. Enhanced Group Classification

TL;DR

This paper performs a comprehensive group classification of a class of nonlinear diffusion-convection equations with variable coefficients, highlighting the importance of extended equivalence groups and gauge transformations for simplifying the classification process.

Contribution

It provides the first complete extended group classification for (1+1)-dimensional nonlinear diffusion-convection equations with variable coefficients, emphasizing the role of extended equivalence groups.

Findings

Constructed the usual and extended equivalence groups.

Identified the structure of non-local gauge equivalence transformations.

Carried out the complete group classification using extended equivalence groups.

Abstract

We discuss the classical statement of group classification problem and some its extensions in the general case. After that, we carry out the complete extended group classification for a class of (1+1)-dimensional nonlinear diffusion--convection equations with coefficients depending on the space variable. At first, we construct the usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements. The extended equivalence group has interesting structure since it contains a non-trivial subgroup of non-local gauge equivalence transformations. The complete group classification of the class under consideration is carried out with respect to the extended equivalence group and with respect to the set of all point transformations. Usage of extended equivalence and correct choice of gauges of arbitrary elements play the major role for simple and clear formulation of the final results. The set of admissible transformations of this class is preliminary investigated.