A refined invariant subspace method and applications to evolution equations

TL;DR

This paper refines the invariant subspace method to generate a broader range of exact solutions for evolution equations, using linear ODE solution subspaces as invariant subspaces, demonstrated through nonlinear dissipative systems.

Contribution

It introduces a refined invariant subspace approach that unifies and diversifies solution methods for evolution equations, with applications to nonlinear dissipative systems.

Findings

Identified invariant subspaces linked to linear ODEs for evolution equations

Derived exact solutions with generalized separated variables

Applied method to a two-component nonlinear dissipative system

Abstract

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations was analyzed to shed light on the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.