Opening the Maslov Box for Traveling Waves in Skew-Gradient Systems: Counting Eigenvalues and Proving (In)stability

We obtain here geometric insight into the stability of traveling pulses for reaction-diffusion equations with skew-gradient structure. For such systems, a Maslov index of the traveling wave can be defined and related to the eigenvalue equation for the linearization L about the wave. We prove two main results about this index. First, for general skew-gradient systems, it is shown that the Maslov index gives a lower bound on the number of real, unstable eigenvalues of L. Second, we show how the Maslov index gives an exact count of all unstable eigenvalues for fast traveling waves in a FitzHugh-Nagumo system. The latter proof involves the Evans function and reveals a new geometric way of understanding algebraic multiplicity of eigenvalues.