Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction

TL;DR

This paper introduces a new symmetry-based method called group foliation reduction for finding exact solutions to nonlinear PDEs, demonstrated on a reaction-diffusion equation, yielding explicit invariant and non-invariant solutions.

Contribution

It presents a novel symmetry method utilizing group foliation reduction to derive explicit solutions of nonlinear PDEs, including non-invariant solutions, which were previously difficult to obtain.

Findings

Derived explicit solutions including similarity and travelling-wave solutions

Obtained non-invariant solutions not admitted by point symmetries

Demonstrated the method on a multi-dimensional reaction-diffusion equation

Abstract

A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this group-foliation reduction method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.