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.
