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

Erneux 9625843调查者研究了一系列与奇摄动和分支问题相关的数学问题。该项目分为三个截然不同的部分。答：慢通过问题：他分析了固体激光器通过谐振的慢通过和锁定半导体激光器的慢通过。这两个问题都是由目前的实验工作引起的。B.激光不稳定性：众所周知，许多实用激光器表现出衰减或持续的脉动振荡。他研究了一系列具体问题，并发展了描述这些问题的渐近方法。C.生物和医学问题：他研究了三个不同的问题，其中一个涉及耦合化学反应，一个是爆发振荡的模型，另一个是一类模拟药物控制释放的移动边界问题。常见的困难是标准准稳态近似的失败。研究人员研究一系列实际问题，这些问题很难、很长或很昂贵，用传统的实验方法来研究。这是许多商业使用的激光器表现出不希望看到的不稳定性的情况，也是允许长时间释放但物理上了解很少的新制药设备的情况。这些问题中的许多都是通过数值研究的数学模型来描述的。这些数值模拟可能很长，因为模型依赖于几个参数，或者因为数值困难。该项目的主要目标是确定这些问题的解析解，然后可能减少或指导数值模拟。这个项目中描述的每个具体问题(激光-生物学)都是由目前与实验小组或生物物理学家的合作推动的。

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.