Title: Instability of pulses in gradient reaction-diffusion systems: A symplectic approach,
Abstract: In a scalar reaction-diffusion equation, it is known that the stability of a
steady state can be determined from the Maslov index, a topological invariant
that counts the state's critical points. In particular, this implies that pulse
solutions are unstable. We extend this picture to pulses in reaction-diffusion
systems with gradient nonlinearity. In particular, we associate a Maslov index
to any asymptotically constant state, generalizing existing definitions of the
Maslov index for homoclinic orbits. It is shown that this index equals the
number of unstable eigenvalues for the linearized evolution equation. Finally,
we use a symmetry argument to show that any pulse solution must have nonzero
Maslov index, and hence be unstable.