Particular solutions to multidimensional PDEs represented in the form of one-dimensional flow

TL;DR

This paper introduces an algorithm that reduces multidimensional nonlinear PDEs, representable as one-dimensional flows, to lower-dimensional PDEs or ODEs, enabling explicit solutions and spectral analysis.

Contribution

It presents a novel reduction method transforming complex multidimensional PDEs into simpler forms, facilitating explicit solutions and spectral parameter integration.

Findings

Reduction of (M+1)-dimensional PDEs to M-dimensional PDEs or ODEs

Explicit solutions for specific nonlinear PDEs demonstrated

Spectral parameters introduced for linear spectral equations

Abstract

We represent an algorithm reducing the $(M+1)$-dimensional nonlinear partial differential equation (PDE) representable in the form of one-dimensional flow $u_t + w_{x_1}(u,u_{x},u_{xx},\dots)=0$, (where $w$ is an arbitrary local function of $u$ and its $x_i$-derivatives, $i=1,\dots,M$) to the family of $M$-dimensional nonlinear PDEs $F(u,w)=0$, where $F$ is general (or particular) solution of a certain second order two-dimensional nonlinear PDE. Particularly, the $M$-dimensional PDE might be an ODE which, in some cases, may be integrated yielding the explicite solutions to the original ($M+1$)-dimensional PDE. Moreover, the spectral parameter may be introduced into the function $F$ which yields a linear spectral equation associated with the original PDE. Simplest examples of nonlinear PDEs with explicite solutions are given.