A Floer homology approach to travelling waves in reaction-diffusion equations on cylinders

TL;DR

This paper introduces a new Floer homology invariant for analyzing bounded solutions of reaction-diffusion PDEs on cylinders, capable of handling minimal restrictions on nonlinearity and providing bounds on solution counts.

Contribution

The paper develops a novel Floer homology framework for reaction-diffusion equations with minimal nonlinearity restrictions, enabling computation and solution counting.

Findings

Homology is invariant under certain nonlinear perturbations.

Provides lower bounds on the number of bounded solutions.

Applicable to PDEs with polynomial growth nonlinearities.

Abstract

We develop a new homological invariant for the dynamics of the bounded solutions to the travelling wave PDE \[ \left\{ \begin{array}{l l} \partial_t^2 u - c \partial_t u + \Delta u + f(x,u) = 0 \qquad & t \in \mathbf{R},\; x \in \Omega, \newline B(u) = 0 & t \in \mathbf{R},\; x \in \partial \Omega, \end{array} \right. \] where $c \neq 0$, $\Omega \subset \mathbf{R}^d$ is a bounded domain, $\Delta$ is the Laplacian on $\Omega$, and $B$ denotes Dirichlet, Neumann, or periodic boundary data. Restrictions on the nonlinearity $f$ are kept to a minimum, for instance, any nonlinearity exhibiting polynomial growth in $u$ can be considered. In particular, the set of bounded solutions of the travelling wave PDE may not be uniformly bounded. Despite this, the homology is invariant under lower order (but not necessarily small) perturbations of the nonlinearity $f$, thus making the homology amenable for computation. Using the new invariant we derive lower bounds on the number of bounded solutions to the travelling wave PDE.