Spectral stability of small-amplitude traveling waves via geometric singular perturbation theory

Abstract

This thesis is concerned with the spectral stability of small-amplitude traveling waves in two different systems: First, in a system of reaction-diffusion equations where the reaction term undergoes a pitchfork bifurcation; second, in a strictly hyperbolic system of viscous conservation laws with a characteristic family that is not genuinely nonlinear.In either case, there exist families of small-amplitude traveling waves. The eigenvalue problem associated with the linearization at the wave is a system of ordinary differential equations depending on two parameters, the amplitude and the spectral value. Suitably scaled, the system reveals a slow-fast structure. Using methods from geometric singular perturbation theory, this will be exploited to thoroughly describe the dynamics of the eigenvalue problem in the zero-amplitude limit. I will prove that the eigenvalue problem converges to the well-understood eigenvalue problem associated with a traveling wave of a certain scalar equation.The proofs rely on concepts from dynamical system theory, most notably on invariant manifold theory and geometric singular perturbation theory.