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

研究者研究了一系列与奇异摄动和分岔问题有关的数学问题。该项目分为三个不同的部分。a .慢通道问题：他分析了固体激光器通过共振的慢通道和半导体激光器通过锁定的慢通道。这两个问题都是由当前的实验工作引起的。B.激光的不稳定性：已知许多实用激光器表现出阻尼或持续的脉动振荡。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：他研究三个不同的问题，一个涉及耦合化学反应，一个爆发振荡模型，以及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。研究者研究了一系列实际问题，这些问题用传统的实验方法来研究是困难的、漫长的或昂贵的。这是许多商业上使用的激光器的情况，它们表现出不期望的不稳定性，以及新的制药设备，它们允许长时间释放，但对物理知之甚少。这些问题中的许多都是用数学模型来描述的，这些模型是用数值方法研究的。这些数值模拟可能会很长，因为模型依赖于几个参数，或者因为数值上的困难。该项目的主要目标是确定这些问题的解析解，从而减少或指导数值模拟。本项目（激光-生物学）中描述的每个具体问题都是由当前与实验组或生物物理学家的合作所激发的。

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.