Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation

TL;DR

This paper classifies nonclassical symmetries of the fast diffusion equation, introduces new potential nonclassical symmetries, and connects these with known solutions, enhancing understanding of the equation's symmetry structure.

Contribution

It provides a complete classification of potential nonclassical symmetries for the fast diffusion equation and links these to known solutions, expanding the symmetry analysis framework.

Findings

New classes of potential nonclassical symmetries identified

Existing non-Lie solutions are extended and characterized

Connections established between nonclassical and potential symmetries

Abstract

The fast diffusion equation $u_t=(u^{-1}u_x)_x$ is investigated from the symmetry point of view in development of the paper by Gandarias [Phys. Lett. A 286 (2001) 153-160]. After studying equivalence of nonclassical symmetries with respect to a transformation group, we completely classify the nonclassical symmetries of the corresponding potential equation. As a result, new wide classes of potential nonclassical symmetries of the fast diffusion equation are obtained. The set of known exact non-Lie solutions are supplemented with the similar ones. It is shown that all known non-Lie solutions of the fast diffusion equation are exhausted by ones which can be constructed in a regular way with the above potential nonclassical symmetries. Connection between classes of nonclassical and potential nonclassical symmetries of the fast diffusion equation is found.