A Floer Homology Approach to Traveling Waves in Reaction-Diffusion Equations on Cylinders

Abstract

Traveling waves form a prominent feature in the dynamics of scalar reaction-diffusion equations on unbounded cylinders. The traveling waves can be identified with the bounded solutions of the elliptic PDE $\begin{cases} \partial_t^2 u - c \partial_t u + \Delta u + f(x,u) = 0, \qquad & t \in \mathbf{R},\; x \in \Omega, \\ B(u) = 0, & t \in \mathbf{R},\; x \in \partial\Omega, \end{cases}$ where $c \neq 0$ is the wave speed, $\Omega \subset \mathbf{R}^d$ is a bounded domain, $\Delta$ is the Laplacian on $\Omega$, and $B$ denotes Dirichlet, Neumann, or periodic boundary data. We develop a new homological invariant for the dynamics of the bounded solutions of the above elliptic PDE. 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 traveling 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 of our PDE, thus obtaining existence and multiplicity results for traveling wave solutions of reaction-diffusion equations on unbounded cylinders.