Conservation laws and hierarchies of potential symmetries for certain diffusion equations

TL;DR

This paper clarifies the nature of hidden potential symmetries in diffusion equations, classifies conservation laws, and systematically derives potential symmetries and invariant solutions, including for linearizable equations.

Contribution

It demonstrates that certain hidden symmetries are actually simple potential symmetries and provides a comprehensive classification and derivation method for potential symmetries in diffusion equations.

Findings

Hidden potential symmetries are ordinary potential symmetries.

Complete classification of conservation laws for porous medium equations.

Infinite-dimensional algebras of potential symmetries constructed.

Abstract

We show that the so-called hidden potential symmetries considered in a recent paper [Gandarias M., Physica A, 2008, V.387, 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.