Statistical Modeling and Predictability of Nonlinear Dispersive Waves

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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.