Instability of pulses in gradient reaction-diffusion systems: a symplectic approach
Beck, Margaret A.
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Citation (published version)MA Beck, Yuri Latushkin, Graham Cox, Kelly Mcquighan, Christopher Jones, Alim Sukhtayev. "Instability of pulses in gradient reaction-diffusion systems: a symplectic approach." Philosophical Transactions A: Mathematical, Physical and Engineering Sciences,
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.