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

小行星9400032 该项目的重点是发展多尺度奇异摄动技术和非线性WKB方法，以研究高频振荡与非线性色散波的相互作用。一个特别的重点将放在快速强迫振幅方程中产生的流体动力稳定性理论。一个总的目标将是构造显式近似解，以渐近展开的形式显示高频和低频波的相互作用。具体来说，我们将研究一个快速强制金斯堡-朗道模型的相关过渡到湍流问题，和扎哈罗夫系统，双曲色散系统建模的快速声波与色散等离子体的相互作用。该提案中描述的工作的一个主要目标是了解振幅方程层次结构中的高阶项如何影响低阶动力学波相互作用。一般来说，高阶项可以将高频振荡馈送到低阶系统中，产生重要的长时间效应，类似于天体力学背景下众所周知的长期行为。待开发的方法将提供一个有用的方式来跟踪外部产生的高频振荡的影响，不断发展的非线性色散波。

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.