Nonlinear Waves in Kinetic and Macroscopic Models

This project is concerned with existence, stability, and numerical simulation of travelling wave solutions of certain model equations of applied mathematics. Travelling waves are special solutions with 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 on the macroscopic quantities (e.g. pressure, density, temperature) that propagates through the medium. An everyday life example are sonic booms produced by aircrafts. These changes are reflected mathematically in the form of discontinuous solutions or shocks. We consider models for which travelling wave solutions represent a continuous (internal) structure of shocks. In this context we are interested in two types of models: 1. A class of models that give a description at a microscopic level (kinetic description). In this case we concentrate on models describing charge transport in semiconductors and plasmas, including an example from the theory of microwave generators. 2. Another class of models, which are derived at the continuum level (macroscopic models) to account for the physical phenomena relevant for the structure of shocks. In this case, we concentrate on a model describing stratified flows. For the first class of problems a first step in the mathematical treatment is to proof 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. For the second problem we shall start with physically meaningful limiting cases of the problem, and perform perturbation arguments in order to get more general results. In both cases, theoretical investigations might be inspired and complemented with numerical experiments.

This project has been 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 considered (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) uses a refined description. Our special interest was 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 was a proof of existence of small amplitude travelling wave solutions. Stability of these waves has been checked by trying to find solution components strongly growing under the influence of perturbations. These theoretical investigations have been inspired and complemented by computer experiments for the numerical approximation of travelling wave solutions.