Group Classification of Generalised Eikonal Equations

TL;DR

This paper develops a comprehensive group classification method for a broad class of nonlinear first-order equations, generalizing eikonal and Hamilton-Jacobi equations, and identifies equations with extensive symmetry groups.

Contribution

It introduces a new approach to classify symmetries of nonlinear PDEs with multiple variables, providing a complete list of equations with symmetry extensions and constructing new equations with large symmetry groups.

Findings

Complete classification of equations with symmetry extensions.

Identification of equivalence groups and algebraic properties.

Construction of new equations with wide symmetry groups.

Abstract

A new approach to the problem of group classification is applied to the class of first-order non-linear equations of the form $u_a u_a=F(t,u,u_t)$. It allowed complete solution of the group classification problem for a class of equations for functions depending on multiple independent variables, where highest derivatives enter nonlinearly. Equivalence groups of the class under consideration and algebraic properties of the symmetry algebra are studied. The class of equations considered presents generalisation of the eikonal and Hamilton-Jacobi equations. The paper contains the list of all non-equivalent equations from this class with symmetry extensions, and proofs of such non- equivalence. New first order non-linear equations possessing wide symmetry groups were constructed.