Mathematical Sciences: Asymptotics of Forced Amplitude Equations from Hydrodynamic Stability Theory

该项目重点发展多尺度奇异摄动技术和非线性WKB方法，研究高频振荡与非线性色散波的相互作用。将特别强调在水动力稳定性理论中出现的快速强迫振幅方程。一般目标将是构造显式近似解，以渐近展开的形式显示高频波和低频波的相互作用。具体来说，我们将研究一个与湍流过渡问题相关的快速强迫金兹堡-朗道模型，以及Zakharov系统，一个模拟快速声波与色散等离子体相互作用的双曲色散系统。提案中描述的工作的主要目标是了解振幅方程层次中的高阶项如何影响低阶动态波相互作用。一般来说，高阶项可以为低阶系统提供高频振荡，产生重要的长时间效应，类似于天体力学中众所周知的长期行为。待开发的方法将提供一种有用的方法来跟踪外部产生的高频振荡对演化的非线性色散波的影响。

9400032 Newton The project focuses on the development of multi-scale singular perturbation techniques and nonlinear WKB methods to study the interaction of high frequency oscillations with nonlinear dispersive waves. A particular emphasis will be placed on rapidly forced amplitude equations arising in hydrodynamic stability theory. A general goal will be to construct explicit approximate solutions showing interactions of high and low frequency waves in the form of asymptotic expansions. Specifically, we will study a rapidly forced Ginzburg-Landau model of relevance to transition to turbulence problems, and the Zakharov system, a hyperbolic- dispersive system modeling the interaction of fast acoustic waves with a dispersive plasma. A main goal of the work described in the proposal is to understand how higher order terms in an amplitude equation hierarchy can affect lower order dynamical wave interactions. In general, the higher order terms can feed high frequency oscillations into the lower order system creating important long time effects akin to the secular behavior well known in the context of celestial mechanics. The methods to be developed will give a useful way of tracking the influence of externally generated high frequency oscillations on evolving nonlinear dispersive waves.