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

在本项目中，首席研究员将继续研究流体动力学中振幅方程时变解的稳定性和分岔问题。特别地，他将研究来自流体动力稳定性理论的金兹堡-朗道方程和来自弱等离子体湍流理论的Zakharov方程组。此外，首席研究员将考虑这两组方程的非线性薛定谔极限。攻击的方法将是一种分析和数值相结合的方法，寻求推导出易于处理的方程的渐近版本，这些方程包含正在建模的现象的基本物理特性。日常生活中的许多自然现象都涉及数学家和物理学家所说的“稳定性和分岔”现象。也就是说，一个系统将或多或少地以一种稳定的运行模式继续下去，直到它受到足够的扰动，从而从旧的行为中产生一种新的行为。当这种情况发生时，人们会说“分叉”发生了，然后人们会对新解的“稳定性”感兴趣。也就是说，新的行为模式会存在多久，需要多大的扰动才能将系统踢入新的行为模式？在这个项目中，首席研究员将通过分析和数值方法研究等离子体和湍流建模中出现的问题。

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.