Reduction of multidimensional non-linear d'Alembert equations to two-dimensional equations: ansatzes, compatibility of reduction conditions

TL;DR

This paper investigates the reduction of multidimensional nonlinear wave equations, specifically d'Alembert equations, to two-dimensional forms using ansatzes, and establishes compatibility conditions for these reductions.

Contribution

It provides necessary compatibility conditions for reducing multidimensional wave equations and classifies possible reduced equations and ansatzes.

Findings

Derived necessary conditions for reduction compatibility.

Described possible types of reduced equations and ansatzes.

Reviewed literature on compatibility and solutions of nonlinear d'Alembert equations.

Abstract

We study conditions of reduction of multidimensional wave equations - a system of d'Alembert and Hamilton equations. Necessary conditions for compatibility of such reduction conditions are proved. Possible types of the reduced equations and ansatzes are described. We also provide a brief review of the literature with respect to compatibility of the system of d'Alembert and Hamilton equations and construction of solutions for the nonlinear d'Alembert equation.