Implicit solutions to some Lorentz invariant non-linear equations revisited

TL;DR

This paper revisits an implicit solution to a Lorentz invariant nonlinear equation, deriving it from a modern form and demonstrating its applicability to linear wave equations.

Contribution

It revives and derives an implicit solution to the Universal Field Equation using a linear ansatz, extending its applicability to linear wave equations.

Findings

Revived implicit solution to the Universal Field Equation.

Derived solutions applicable to linear wave equations.

Connected historical solutions with modern PDE methods.

Abstract

An implicit solution to the vanishing of the so-called Universal Field Equation, or Bordered Hessian, which dates at least as far back as 1935 \cite{chaundy} is revived, and derived from a much later form of the solution. A linear ansatz for an implicit solution of second order partial differential equations, previously shown to have wide applicability \cite{fai} is at the heart of the Chaundy solution, and is shown to yield solutions even to the linear wave equation.