Modul

The module examination takes place in form of an oral exam of about 30 minutes. Please see under "Modulnote" for more information about the bonus regulation.

After successful completion of this module students:

• can explain the significance of traveling waves and their dynamic stability;
• know basic methods to study the existence of traveling waves;
• outline the main steps in a stability analysis and address potential complications;
• have acquired several mathematical tools to compute or approximate the spectrum;
• master several techniques to derive (in)stability of the wave from spectral information;
• understand how spectrum and stability might depend on the class of perturbations.

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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.

The following background is strongly reommended: Analysis 1-4.

Total workload: 180 hours

Attendance: 60 hours

• lectures, problem classes, and examination 

Self-studies: 120 hours

• follow-up and deepening of the course content,
• work on problem sheets,
• literature study and internet research relating to the course content,
• preparation for the module examination