# Instability of pulses in gradient reaction-diffusion systems: A symplectic approach

## TL;DR

This paper extends the use of the Maslov index to reaction-diffusion systems with gradient nonlinearity, demonstrating that pulse solutions are inherently unstable due to their nonzero Maslov index.

## Contribution

It generalizes the Maslov index to reaction-diffusion systems, linking it to stability and proving pulse instability via symmetry arguments.

## Key findings

- Maslov index equals the number of unstable eigenvalues.
- Pulse solutions in these systems are always unstable.
- The approach generalizes existing topological stability methods.

## 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.