  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.
