### Traveling Waves (Winter Semester 2022/23)

- Lecturer: Dr. Björn de Rijk
- Classes: Lecture (0108100), Problem class (0108110)
- Weekly hours: 3+1

Schedule | ||
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Lecture: | Tuesday 8:10-9:40 | 20.30 SR 2.66 |

Wednesday 15:45-17:15 (every 2nd week) | 20.30 SR 2.66 | |

Problem class: | Wednesday 15:45-17:15 (every 2nd week) | 20.30 SR 2.66 |

Lecturers | ||
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Lecturer, Problem classes | Dr. Björn de Rijk | |

Office hours: Office hours: by appointment | ||

Room -1.019 Kollegiengebäude Mathematik (20.30) | ||

Email: bjoern.rijk@kit.edu |

# Contents

Traveling waves are solutions to nonlinear partial differential equations (PDEs) that propagate over time with a fixed speed without changing their profiles. These special solutions arise in many applied problems where they model, for instance, water waves, nerve impulses in axons or light in optical fibers. Therefore, their existence and the naturally associated question of their dynamic stability is of interest, because only those waves which are stable can be observed in practice.

The first step in the stability analysis is to linearize the underlying PDE about the wave and compute the associated spectrum, which is in general a nontrivial task. To approximate spectra associated with various waves, such as fronts, pulses and periodic wave trains, we introduce the following tools:

- Sturm-Liouville theory
- exponential dichotomies
- Fredholm theory
- the Evans function
- parity arguments
- essential spectrum, point spectrum and absolute spectrum
- exponential weights

The next step is to derive useful bounds on the linear solution operator, or semigroup, based on the spectral information. A complicating factor is that any non-constant traveling wave possesses spectrum up to the imaginary axis. For various dissipative PDEs, such as reaction-diffusion systems, we employ the bounds on the linear solution operator to close a nonlinear argument via iterative estimates on the Duhamel formula. For traveling waves in Hamiltonian PDEs, such as the NLS or KdV equation, we describe a different route towards stability based on the variational arguments of Grillakis, Shatah and Strauss.

# Examination

The module examination at the end of the semester takes place in the form of an oral exam of about 30 minutes. After *passing* the oral exam, the final grade is min{0.7*X* + 0.3*Y*, *X*}, where *X* is the grade for the oral exam and *Y* is the grade obtained by working out and presenting a model problem during one of the exercise classes.

# References

We will follow the book: Kapitula, Todd; Promislow, Keith. Spectral and dynamical stability of nonlinear waves. Applied Mathematical Sciences, 185. Springer, New York, 2013. (available as e-book within the KIT university network)