Opening the Maslov Box for Traveling Waves in Skew-Gradient Systems

Abstract

We obtain 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.

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References in corpus (2)

• The Maslov and Morse indices for Schrödinger operators on $\mathbb{R}$
• A Stability Index for Traveling Waves in Activator-Inhibitor Systems

Cited by in corpus (4)

• On the Existence and Stability of Fast Traveling Waves in a Doubly-Diffusive FitzHugh-Nagumo System
• Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions
• Travelling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise
• Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise