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

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

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WÄCHTLER, Johannes, 2012. Spectral stability of small-amplitude traveling waves via geometric singular perturbation theory [Dissertation]. Konstanz: University of Konstanz

@phdthesis{Wachtler2012Spect-19951, title={Spectral stability of small-amplitude traveling waves via geometric singular perturbation theory}, year={2012}, author={Wächtler, Johannes}, address={Konstanz}, school={Universität Konstanz} }

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.<br /><br /><br /><br />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.<br /><br /><br /><br />The proofs rely on concepts from dynamical system theory, most notably on invariant manifold theory and geometric singular perturbation theory. Wächtler, Johannes eng Wächtler, Johannes 2012-08-03T08:49:12Z Spectral stability of small-amplitude traveling waves via geometric singular perturbation theory 2012-08-03T08:49:12Z 2012 terms-of-use

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