Nonlinear waves in kinetic and macroscopic models

This project is concerned with questions of existence, stability, and numerical simulation of travelling wave solutions of certain model equations of applied mathematics. Based on an intuitive notion of waves, travelling waves are spatially localized structures moving with constant velocity. Although travelling waves are very special solutions, in some situations they can be used as building blocks of more general phenomena. Some of the model equations we consider (so called macroscopic models) describe a continuous medium (such as air) by macroscopic quantities like density, average velocity and temperature. The second class of models (so called kinetic models) use a refined description. Our special interest is in macroscopic models where certain phenomena (such as a sonic boom) can only be described by jumps of certain quantities, whereas a kinetic description provides a continuous resolution of the structure in the form of a so called shock profile. On the other hand, the disadvantage of kinetic descriptions is their much more complicated theoretical and numerical treatment. Other model problems are mathematical descriptions of charge transport in semiconductors and plasmas, including an example from the theory of microwave generators. The first step in the mathematical treatment is a proof of existence of small amplitude travelling wave solutions. Stability of these waves is checked by trying to find solution components strongly growing under the influence of perturbations. These theoretical investigations will be inspired and complemented by computer experiments for the numerical approximation of travelling wave solutions.

This project dealt with existence, stability, and numerical simulation of traveling wave solutions of certain equatuions of applied mathematics. Travelling waves are special solutions which keep their profile and move with constant speed. They are essential in the explanation of some general phenomena. For example, in continuous media (e.g. gases) disturbances might cause an instantaneous change of macroscopic quantities (e.g. pressure, density, temperature) that propagates through the medium. An everyday life example are sonic booms produced by aircrafts. These abrupt changes are reflected mathematically in the form of discontinuous solution. Physically more refined mathematical models as the kinetic models considered here, are able to resolve the continuous (internal) structure of shocks in the form of continuous traveling waves. This model class describes the dynamics of gases as well as charge transport in semiconductors and plasmas. The main results o the project are existence proofs of these traveling wave solutions and proofs of their dynamic stability. Travelling wave solutions also appear in other contexts, such as in equations describing chemical reactions and combustion processes. In such examples, they might explain general phenomena, such as the uniform propagation of chemical compounds or of flames. In this project, an example from the theory of microwave generators has been considered, whose performance relies on traveling charge waves through a semiconductor crystal.