FWF Project No. P18367

Nonlinear Waves in Kinetic and Macroscopic Models


Principle Investigator: PostDocs: PreDoc:
Funding period: April 2006 - March 2009



Scientific aims

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.