General Solution for a Coupled System of Eikonal Equations in Two Space Variables

TL;DR

This paper derives a general solution for a coupled system of eikonal equations in two spatial variables, using hodograph and contact transformations, with implications for mathematical physics applications.

Contribution

It introduces a novel method to solve a coupled system of eikonal equations, expanding the analytical tools available for such nonlinear PDE systems.

Findings

General solution for the coupled eikonal system obtained

Method applicable to eikonal-Hamilton-Jacobi systems

Potential applications in mathematical physics

Abstract

A general solution for a coupled system of eikonal equations u_\mu u_\mu = 0, v_\mu v_\mu = 0, u_\mu v_\mu = 1 is presented, where lower indices designate derivatives, \mu = 0, 1, 2 and summation is implied over the repeated indices. This solution is of interest by itself due to wide applications of the eikonal equations, but the system considered also appears to be part of the reduction conditions for many equations of mathematical physics. We describe in detail the procedure that allowed obtaining of the general solution using hodograph and contact transformations of the initial system, however, we omit here special case when the system is equivalent to a system for one space dimension or to a system for one dependent function. The procedure used allowed also obtaining of the general solution for a coupled system of the eikonal and Hamilton-Jacobi equation.