Wave Turbulence: A Wealth of Applications and a Rich Paradigm for Turbulent Systems

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Abstract: DMS-0404577A Newell and V ZakharovWave Turbulence: A Wealth of Applications and a Rich Paradigm for Turbulent SystemsWave turbulence is the study of the statistical behavior of solutions of field equations, usually conservative and Hamiltonian, describing weakly nonlinear and dispersive wave systems in the presence of sources and sinks. It has many applications, from ocean waves to magnetohydrodynamics and optics. But it is not only of interest because of applications. It provides a nontrivial tool for examining the long time solution behavior of the underlying Hamiltonian partial di_erential equations. It is extraordinarily rich in that, save for very rare circumstances, the theory is not valid for all scales. We have recently developed an understanding for how to predict the breakdown ranges in whichrandomly occurring, robust and fully nonlinear coherent structures occur. These lead to anomalous and intermittent behaviors whose signatures can be seen dramatically in high order moments and sometimes even in low order ones. It also leads to large deviations from Gaussian statistics. Because in several cases it is possible to identify and understand the reasons for the emergence of the randomly occurring structures, wave turbulence provides a nontrivial paradigm for investigating the guiding principles governing turbulent and nonequilibrium systems in general.Wave turbulence has a wealth of applications and is also a most useful paradigm for the study of nonequilibrium systems. Therefore, it is an excellent training tool for students and young scientists because many solutions and bulk properties are calculable, thus helping those new to the field to gain a concrete understanding of central issues. The two PI's have been linked with formidable stables of young researchers throughout their professional lives and have produced more than 20 Ph.D's who are actively contributing to the field of nonlinear science. Further, we include in this proposal several concrete actions we plan to take in order to facilitate the training of new researchers, to collaborate with teams throughout the world and to disseminate new results.