Statistical Modeling and Predictability of Nonlinear Dispersive Waves

数学科学：非线性色散波的统计建模和可预测性摘要[0206679]本研究的中心主题是空间扩展的多尺度非线性系统，特别是非线性色散波的统计可预测性和有效动力学的发展。多尺度非线性动力学所表现出的复杂行为建模往往需要对大规模、粗粒度动力学进行有效描述。它们的解决需要对系统中存在的所有空间和时间激励进行精确的数学表征。时空混沌的长时间、大尺度动力学的统计特性的量化问题将在一个近积分的设置和一个系统中得到解决，在这个系统中尺度的分离以及不稳定性可以被精确地调谐和控制。一旦获得良好的统计表征，它不仅可以为粗粒度动力学建模提供指导，还可以根据原始全动力学对这些有效模型进行统计校准。有了这些统计见解，研究进一步集中在两种可能情况下的粗粒度动态和不变测度的研究上——即有和没有尺度分离的动态。这些项目还涉及色散波湍流的重要方面：澄清动力学方程的推导及其验证；并详细表征共振条件，通量动力学和空间局域化，相干结构。从蛋白质折叠的分子动力学模拟到大气-海洋耦合动力学的短期气候预测，在现代科学建模问题中可能会出现大量的时空尺度，这些问题的研究对我们的日常生活有很大的影响。这个项目研究了可以用来理解和预测这类系统复杂行为的数学方法。

NSF Award Abstract - DMS-0206679Mathematical Sciences: Statistical modeling and predictability of nonlinear dispersive wavesAbstract0206679 CaiThe central theme of this research is the statistical predictability and the development of effective dynamics for spatially extended, multi-scale nonlinear systems in general, and nonlinear dispersive waves in particular. Modeling complex behavior exhibited by multi-scale nonlinear dynamics often entails an effective description of large scale, coarse-grained dynamics. Their resolution requires a precise mathematical characterization of all spatial and temporal excitations present in systems. The issue of quantification of statistical properties of long-time, large-scale dynamics of spatiotemporal chaos will be addressed in a near-integrable setting and in a system in which the separation of scales, as well as instability, can be precisely tuned and controlled. Once a good statistical characterization is obtained, it can provide not only guidance in modeling coarse-grained dynamics but also statistical calibrations of these effective models against the original full dynamics. With these statistical insights, the research further focuses on the study of coarse-grained dynamics and invariant measures for two possible situations --- namely, dynamics with and without separation of scales. The projects also address important aspects of dispersive wave turbulence: clarification of the derivation of kinetic equations and their validation; and detailed characterization of resonance conditions, flux dynamics, and spatially localized, coherent structures.A multitude of spatiotemporal scales may arise in modeling problems in modern science, ranging from molecular dynamics simulation of protein folding to short term climate prediction for coupled atmosphere-ocean dynamics --- the study of which has great impact on our daily world. This project investigates mathematical methods that can be used to understand and predict the complicated behavior of systems of this type.