Wave Turbulence: Open Challenges and New Opportunities

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NSF Award Abstract - DMS-0072803Mathematical Sciences: Wave Turbulence -- Open Challenges and New OpportunitiesAbstract0072803 ZakharovWave turbulence, the turbulence of a sea of weakly-coupled dispersive wave trainsdriven far from equilibrium by sources and sinks, is ripe for renewed investigation.First, there are many open challenges which need to be addressed: the effectof discreteness in impeding spectral energy redistribution; the general nature ofanisotropic finite flux spectra in wind-driven oceans, in magnetohydrodynamics inthe presence of strong magnetic field, and in sound waves; the role of hidden constantsof the motion; the surprising manner in which finite capacity Kolmogorov spectraare realized; the modification of Fermi-Dirac (and Bose-Einstein) spectra in contextssuch as semiconductor lasing where finite flux effects are important. Second, thesubject is important because it provides tractable models which can be exploitedto gain insight into the phenomenon of intermittency. Because the approximationswhich underlie the applications of wave turbulence theory are generally not valid uniformly in wave number, there are windows of intermittency at both high and low wave number values. At these scales, the system is fully nonlinear and exhibits intermittent bursts of activity associated with coherent but almost singular structures. Our aim is to build a comprehensive theory which combines and connects wave turbulence with the random occurrence of coherent events.This project investigates the long-time statistical behavior of complex solutions to nonlinear field equations arising in a broad family of physical systems. Applications of the research include modeling of: wind-driven ocean waves, weather, laser beams in transparent materials, sound waves, solar wind and stellar atmospheres, and motion of charges in semiconductor lasers. All these systems exhibit turbulent behavior with common features, and the project develops the theory of this common structure. The goal is to gain understanding of the behavior of this family of physical systems as well as to study the general statistical behavior of nonlinear systems in which large fluctuations