Collections > Electronic Theses and Dissertations > A symplectic view of stability for traveling waves in activator-inhibitor systems
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This thesis concerns the stability of traveling pulses for reaction-diffusion equations of skew-gradient (a.k.a activator-inhibitor) type. The centerpiece of this investigation is a homotopy invariant called the Maslov index which is assigned to curves of Lagrangian planes. The Maslov index has been used in recent years to count positive eigenvalues for self-adjoint Schrodinger operators. Such operators arise, for instance, from linearizing a gradient reaction-diffusion equation about a steady state. In that case, positive eigenvalues correspond to unstable modes. In this work, we focus on two aspects of the Maslov index as a tool in the stability analysis of nonlinear waves. First, we show why and how the Maslov index is useful for traveling pulses in skew-gradient systems, for which the associated linear operator is not self-adjoint. This leads naturally to a discussion of the famous Evans function, the classic eigenvalue-hunting tool for steady states of semilinear parabolic equations. A major component of this work is unifying the Evans function theory with that of the Maslov index. Second, we address the issue of calculating the Maslov index, which is intimately tied to its utility. The key insight is that the relevant curve of Lagrangian planes is everywhere tangent to an invariant manifold for the traveling wave ODE. The Maslov index is then encoded in the twisting of this manifold as the wave moves through phase space. We carry out the calculation for fast traveling pulses in a doubly-diffusive FitzHugh-Nagumo system. The calculation is made possible by the timescale separation of this system, which allows us to track the invariant manifold of interest using techniques from geometric singular perturbation theory. Combining the calculation with the stability framework established in the first part, we conclude that the pulses are stable.