Eikonal Equation 0.1

TL;DR

This paper reviews symmetry properties of eikonal equations, explores their solutions and reductions, and introduces new results including a general solution for any number of spatial variables, serving as both reference and manual.

Contribution

It provides a comprehensive review of symmetries and solutions of eikonal equations, including new results like the general solution for arbitrary spatial dimensions.

Findings

Analysis of symmetry-related solutions of eikonal equations

Derivation of a general solution for any number of space variables

Detailed procedure using hodograph and contact transformations

Abstract

We provide a review of some symmetry-related literature on the eikonal equations $u_\mu u_\mu =0$,$u_\mu u_\mu =1$, where lower indices at dependent variables designate derivatives, $\mu=0,1,2,..,n$ and summation is implied over the repeated indices. We will consider general solutions and symmetries of these equations, and relations of these equations with the reduction of higher-order PDE. Some new results that were needed for the comprehensive presentation are also adduced. In particular, we will also consider discrete symmetries of the eikonal equations equivalence classes of solutions and relations of the symmetry solutions and the general solutions. We describe in detail the procedure that allowed obtaining of the general solution using hodograph and contact transformations of the initial equations. Some new results by the author that were needed for comprehensive presentation are also adduced - in particular, a general solution of the eikonal equation for an arbitrary number of space variables. The paper is intended both as reference material and as a manual.