## Projektseminar (Functional Analysis): Stability of Nonlinear Waves

Wintersemester 2016/17

Time and Place

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SE 2 std. | Tue 10:15-11:45 | SR12 | 4.10 |

Topics

The stability of linear and nonlinear waves has a fascinating history.
In this seminar we would like to discuss basic approaches to the problem of stability (spectral, orbital and asymptotic) of solitary waves.
Our main aim is to look at a circle of ideas and the mathematical machinery that has been built to investigate the stability problem.

Presentations

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08.11 | Travelling Wave solutions | Gerald Teschl | [Pava] Chapter 3 |

15.11 | Scalar viscous conservation laws or NLS-type equations I | Markus Sobotnik | [K-P] Sect 4.4 or 4.5 |

22.11 | Scalar viscous conservation laws or NLS-type equations II | Markus Sobotnik | [K-P] Sect 4.4 or 4.5 |

29.11 | The gKdV equations I | Markus Holzleitner | [K-P] Sect 5.2.1 |

13.12 | The gKdV equations II | Markus Holzleitner | [K-P] Sect 5.2.1 |

17.01 | General Orbital Stability (+Exercises) | Noema Nicolussi | [K-P] Sect 5.2.2 |

24.01 | Eigenvalues of Constrained Self-Adjoint Operators | Leopold Veselka | [K-P] Sect 5.3 |

References:

- M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal.
**74**, 160-197 (1987); II,**94**, 308-348 (1990). - T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer, New York, 2013.
- J. A. Pava, Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Amer. Math. Soc., Providence, 2009.
- A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys.
**133**, 119-146 (1990). - A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math.
**136**, 9-74 (1999).

Course assessment

Preparation and presentation of a chosen topic. Active participation during the seminar.

Audience

Majors in Mathematics (master program, code MANS), Physics, ...

Auf Ihr Kommen freuen sich Aleksey Kostenko und Gerald Teschl