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

在这个项目中，首席研究员将继续研究与流体动力学中产生的振幅方程的依赖于时间的解的稳定性和分叉有关的几个问题。特别是，他将从流体动力学稳定性理论研究金兹堡-朗道方程，从弱等离子体湍流理论研究扎哈罗夫方程组。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，寻求推导出方程的渐近版本，这些方程是易于处理的，并且包含被建模现象的基本物理。日常生活中的许多自然现象都涉及到数学家和物理学家所说的“稳定性和分叉”现象。也就是说，一个系统将或多或少地以稳定的运行模式继续运行，直到它被充分扰动到从旧的行为中产生新类型的行为的程度。当这种情况发生时，有人说已经发生了“分叉”，然后人们对新解的“稳定性”感兴趣。也就是说，新的行为模式将存在多久，需要多大的扰动才能将系统推向新的行为模式？在这个项目中，首席研究人员将使用解析和数值方法研究等离子体和湍流模型中出现的问题。

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.