Spectral Stability of Small-Amplitude Traveling Waves via Geometric Singular Perturbation Theory

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 φε, ε ∈ (0, ε0], ε0 1, of small-amplitude traveling waves. The eigenvalue problem associated with the linearization at φε is a system of ordinary differential equations depending on two parameters, the amplitude ε and the spectral value κ. 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, in the limit ε → 0, to the well-understood eigenvalue problem associated with a traveling wave φ0 of a certain scalar equation. The profiles φε then inherit the spectral stability from the respective limit profile φ0. The proofs rely on concepts from dynamical system theory, most notably on invariant manifold theory and geometric singular perturbation theory.