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

Type of Publication: | Dissertation |

URI (citable link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-199513 |

Author: | Wächtler, Johannes |

Year of publication: | 2012 |

Summary: |
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. |

Examination date (for dissertations): | May 30, 2012 |

Dissertation note: | Doctoral dissertation, University of Konstanz |

Subject (DDC): | 510 Mathematics |

Keywords: | Traveling waves, spectral stability, geometric singular perturbation theory, Evans function |

Link to License: | In Copyright |

Bibliography of Konstanz: | Yes |

Checksum:
MD5:456dc86c1d7bac80fe4c6ecda0a062de

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} }

JW-final.pdf | 255 |