Representation of solutions of linear PDE's with constant coefficients as superpositions of solutions in lower dimensions

TL;DR

This paper presents a unified method to express solutions of linear PDEs with constant coefficients in high dimensions as superpositions of lower-dimensional solutions, leveraging Fourier integral representations and symmetry properties.

Contribution

It introduces a general approach to decompose high-dimensional PDE solutions into lower-dimensional ones using Fourier integrals and symmetry invariance, applicable to equations like Laplace and wave equations.

Findings

Solutions of high-dimensional Laplace and wave equations can be represented as superpositions of lower-dimensional solutions.

The approach utilizes Fourier integral representations and symmetry invariance under transformations.

Examples demonstrate the method's applicability to classical PDEs in physics and mathematics.

Abstract

A unified approach to the representation of solutions of linear PDE's with constant coefficients in high dimensions in terms of solutions of the same PDE's in lower dimensions is presented. It is based on the observation that if a function of an (N+1)-dimensional variable (N * 2) has a convergent Fourier integral representation, then it may be written as a superposition of rotated plane waves in any number of dimensions lower than N+1. If that function is a solution of a linear PDE with constant coefficients, and, in addition, the equation is invariant under a spatial group of transformations (e.g., rotations, or Lorentz transformations), then the superposition is in terms of solutions of lower- (not only two-) dimensiuonal solutions of the same PDE. As examples, representations of solutions of the Laplace equation in (N+1) dimensions and of the wave equation in (1+N) dimensions (N * 2) as superpositions of solutions in lower dimensions are discussed, and simple consequences are presented.