Symmetry classification of quasi-linear PDE's containing arbitrary functions

TL;DR

This paper develops a method for classifying symmetries of a broad class of quasi-linear PDEs with arbitrary functions, providing conditions for nontrivial Lie symmetries and illustrating with examples from physics.

Contribution

It introduces an easy condition for symmetry existence and offers a geometric characterization for the symmetry-determining system in quasi-linear PDEs.

Findings

Derived a condition for nontrivial Lie symmetries in PDEs with arbitrary functions

Provided a geometric framework for symmetry classification

Applied the method to equations in physics, like magnetohydrodynamics

Abstract

We consider the problem of performing the preliminary "symmetry classification'' of a class of quasi-linear PDE's containing one or more arbitrary functions: we provide an easy condition involving these functions in order that nontrivial Lie point symmetries be admitted, and a "geometrical'' characterization of the relevant system of equations determining these symmetries. Two detailed examples will elucidate the idea and the procedure: the first one concerns a nonlinear Laplace-type equation, the second a generalization of an equation (the Grad-Schl\"uter-Shafranov equation) which is used in magnetohydrodynamics.