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.
