Mathematical Sciences: "Asymptotic & Singular Perturbation Methods for Bifurcation Problems with Applications"

点击翻译按钮获取中文摘要

Erneux 9625843 The investigator studies a series of mathematical problems connected with singular perturbation and bifurcation problems. The project is divided in three distinct parts. A. Slow passage problems: He analyzes the slow passage through resonance in a solid state laser and the slow passage through locking in a semiconductor laser. Both problems are motivated by current experimental work. B. Laser instabilities: Many practical lasers are known to exhibit damped or sustained pulsating oscillations. He investigates a series of specific problems and develop asymptotic methods for their description. C. Biological and medical problems: He works on three different problems, one involving coupled chemical reactions, a model for bursting oscillations, and a class of moving boundary problems modeling the controlled release of pharmaceutical drugs. The common difficulty is the failure of a standard quasi-steady state approximation. The investigator studies a series of practical problems that are difficult, long, or expensive to investigate by using traditional experimental methods. This is the case for many commercially used lasers that exhibit undesired instabilities and for new pharmaceutical devices that allow long releases but are poorly understood physically. Many of these problems are described by mathematical models that are studied numerically. These numerical simulations can be long because the models depend on several parameters or because of numerical difficulties. The main goal of the project is to determine analytical solutions of these problems that may then reduce or guide the numerical simulations. Each specific problem described in this project (lasers - biology) is motivated by current collaborations with experimental groups or with biophysicists.