Nonlinear first order PDEs reducible to autonomous form polynomially homogeneous in the derivatives

TL;DR

This paper establishes a theorem that characterizes when nonlinear first order PDE systems polynomial in derivatives can be transformed into autonomous, polynomially homogeneous systems, leveraging symmetry properties and Lie symmetries.

Contribution

It provides necessary and sufficient conditions for transforming certain nonlinear PDEs into simpler autonomous forms using symmetry analysis.

Findings

The theorem applies to Monge-Ampère systems with constant or variable coefficients.

Constructive proof involves canonical variables from Lie point symmetries.

Conditions enable reduction of complex PDE systems to autonomous, homogeneous forms.

Abstract

It is proved a theorem providing necessary and sufficient conditions enabling one to map a nonlinear system of first order partial differential equations, polynomial in the derivatives, to an equivalent autonomous first order system polynomially homogeneous in the derivatives. The result is intimately related to the symmetry properties of the source system, and the proof, involving the use of the canonical variables associated to the admitted Lie point symmetries, is constructive. First order Monge-Amp\`ere systems, either with constant coefficients or with coefficients depending on the field variables, where the theorem can be successfully applied, are considered.