Mathematical Sciences: Asymptotic and Singular Perturbation Methods for Bifurcation and Moving Boundary Problems with Applications

获得这一奖项后，首席研究员将继续应用奇异摄动型渐近方法研究由反应扩散系统引起的分支问题和移动边界问题。特别是，他将研究所谓的奇异分叉问题，其中分叉参数的微小变化可以在系统中引起大的响应，激光不稳定性，以及涉及药物释放的某些药物问题中移动界面的变化的时间历史。这一建议与分叉现象有关。举个例子，把一把塑料尺子竖立起来，然后推到顶端。如果你不用力推，什么都不会发生。但如果你用力足够大，第一件事就是尺子向左或向右鞠躬。我们称之为分叉：直尺的形状分成了两种可能的形状之一。对标尺施加更大的力会导致偶数个弯曲形状具有越来越多的曲线。首席研究人员将研究从控制激光和体内化学物质释放的方程系统的给定解中分出的解的数量。

With this award the principal investigator will continue his application of asymptotic methods of singular perturbation type to the study of bifurcation problems and moving boundary problems that arise from reaction-diffusion systems. In particular, he will investigate so-called singular bifurcation problems, wherein small changes in the bifurcation parameter can cause large responses in the system, laser instabilities, and the changing time history of the moving interface in certain pharmaceutical problems involving the time-release of medication. This proposal is concerned with bifurcation phenomena. As an illustration, stand a plastic ruler on its end and push on the top end. If you do not push very hard nothing happens. But if you push sufficiently hard the first thing that happens is that the ruler bows, either to the left or to the right. We call this a bifurcation: the configuration of the straight ruler has split (bifurcated) into one of two possible shapes. Applying still more force to the ruler results in an even number of bowed shapes having more and more curves. The principal investigator will study the number of solutions that bifurcate from a given solution of systems of equations that govern lasers and release of chemicals in the body.