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

摘要：DMS-0404577 A纽韦尔和V Zakharov波动湍流：丰富的应用和丰富的湍流系统范例波动湍流是场方程解的统计行为的研究，通常是保守的和哈密顿的，描述了存在源和汇的弱非线性和色散波系统。它有许多应用，从海浪到磁流体力学和光学。但它不仅仅是因为应用程序而感兴趣。它提供了一个非平凡的工具，检查的长期行为的哈密顿偏微分方程。它的丰富之处在于，除了非常罕见的情况之外，这个理论并不是对所有尺度都有效。我们最近已经发展了一种理解，如何预测的崩溃范围内随机发生的，强大的和完全非线性相干结构发生。这些导致异常和间歇性的行为，其签名可以在高阶矩中显着地看到，有时甚至在低阶矩中也可以看到。这也导致了高斯统计的大偏差。由于在某些情况下可以识别和理解随机结构出现的原因，波动湍流为研究一般湍流和非平衡系统的指导原则提供了一个重要的范式，波动湍流有着丰富的应用，也是研究非平衡系统最有用的范式。因此，它是学生和年轻科学家的一个很好的培训工具，因为许多解决方案和整体性质是可计算的，从而帮助那些新的领域，以获得核心问题的具体理解。这两个PI的一直与年轻研究人员的强大马厩在他们的职业生涯，并产生了20多个博士谁是积极贡献的非线性科学领域。此外，我们在本提案中列入了我们计划采取的几项具体行动，以促进新研究人员的培训，与世界各地的团队合作，并传播新成果。

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.