Mathematical Sciences: Asymptotic and Computational Techniques for Amplitude Equations and Weak Turbulence Models

在这个项目中，首席研究员将继续他的 研究了有关稳定性和 振幅方程含时解的分歧 产生于流体动力学。 特别是，他将调查 流体力学理论中的Ginzburg-Landau方程 稳定性理论和Zakharov方程组 弱等离子体湍流理论 此外，校长 研究者将在他们研究中考虑这两组方程， 非线性薛定谔极限 攻击的方法将是 分析和数值相结合的方法，旨在获得 渐近版本的方程是易于处理的， 包含被建模现象的基本物理学。 日常生活中的许多自然现象都涉及到什么 数学家和物理学家称之为“稳定性和分叉” 现象。 也就是说，一个系统将或多或少地在一个 稳定的操作模式，直到它受到足够的扰动， 一种新的行为从旧的行为中产生。 当这种情况发生时，人们说“分叉”已经发生， 于是人们对新解决方案的“稳定性”感兴趣。 也就是说，新的行为模式将存在多久，以及如何 一个很大的扰动会把系统踢进一个 新的行为模式？ 在这个项目中， 将研究这些问题，因为它们出现在模拟 等离子体和湍流的分析和数值方法。

In this project the principal investigator will continue his study of several problems related to the stability and bifurcation of time-dependent solutions of amplitude equations arising in fluid dynamics. In particular, he will investigate the Ginzburg-Landau equation from the theory of hydrodynamic stability theory and the Zakharov system of equations from the theory of weak plasma turbulence. In addition, the principal investigator will consider these two sets of equations in their nonlinear Schroedinger limits. The method of attack will be a combined analytical and numerical approach that seeks to derive asymptotic versions of the equations that are tractable and that contain the essential physics of the phenomena being modelled. Much of the natural phenomena of everyday life involves what mathematicians and physicists call "stability and bifurcation" phenomena. That is, a system will continue more or less in a stable mode of operation until it is perturbed sufficiently to the point where a new type of behavior arises out of the old one. When this happens one says that a "bifurcation" has occurred and one is then interested in the "stability" of the new solution. That is, how long will the new mode of behavior exist and how large of a perturbation will it take to kick the system into a new mode of behavior? In this project the principal investigator will examine such problems as they arise in the modelling of plasmas and turbulence by using analytical and numerical methods.