Asymptotic Regimes of Nonlinear Dispersive Equations

本项目的主要目标是探索对非线性色散偏微分方程的长期行为有重要影响的因素，如非线性薛定谔（NLS），非线性波（NLW），非线性克莱因-戈登（NKG）和扎哈罗夫系统。首先，域几何形状的影响进行了调查的基础流形的体积增长和线性解的衰减之间的关系。这种关系的非线性问题的影响，然后探讨在某些易处理的几何，即欧几里得空间的几何。其次，研究了一类弱非线性色散方程的小数据点散射（渐近线性行为）问题。这些方程的非线性度不足以使经典方法成功。一个重点是由一个潜在的长期代表的不均匀性，以及如何这可能会影响渐近行为。最后，通过研究紧致区域上三次NLS方程的长时间行为，研究了所谓的弱湍流区。在这种情况下，一个新的连续方程推导出的包络的离散傅立叶模式，通过采取适当的大盒限制在弱湍流理论的精神。该方程具有显著的对称性，甚至有显式解，可以用来（通过严格的近似结果）更好地理解原始NLS系统的长时间频率动力学。本项目中考虑的非线性色散方程自然出现在物理学的几个领域（等离子体物理，非线性光学，广义相对论等）。在那里，它们通常用作简化的有效模型。为了使这些模型也能用于工程目的，人们必须理解不均匀性（通常由非欧几里德域几何或势项建模）在与真实的世界建立联系时的影响。这样的研究将有助于设置参数和限制，不仅现有的简化模型的有效性，而且计算机模拟（基于这些模型）在预测实验现象的忠实性。从更理论的角度来看，开始对弱湍流的重要物理理论进行数学上的严格研究，正如提案中所建议的那样，代表了纯数学，应用数学和物理学之间的跨学科合作，在等离子体和流体工程以及海洋和大气科学中有着有益的应用。

The main objective of this project is to explore factors that vitally influence the long-time behavior of nonlinear dispersive PDE like the non-linear Schroedinger (NLS), nonlinear wave (NLW), nonlinear Klein-Gordon (NKG), and Zakharov systems. First, the effect of domain geometry is investigated in terms of a relation between the volume growth of the underlying manifold and the decay of linear solutions. The implications of this relation for the nonlinear problem are then explored in certain tractable geometries, namely quotients of Euclidean spaces. Second, the question of small-data scattering (asymptotically linear behavior) for some weakly nonlinear dispersive equations is considered. These are equations where the nonlinearity is not of high enough degree for classical methods to succeed. One focus is on inhomogeneities represented by a potential term, and how this can affect the asymptotic behavior. Finally, the so-called weakly turbulent regime is investigated by studying the long-time behavior of the cubic NLS equation on a compact domain. In this context, a new continuum equation is derived for the envelope of the discrete Fourier modes by taking a suitable large-box limit in the spirit of weak turbulence theory. This equation turns out to enjoy remarkable symmetries and even explicit solutions, and can be used (via rigorous approximation results) to better understand the long-time frequency dynamics of the original NLS system.The nonlinear dispersive equations considered in this project arise naturally in several areas of physics (plasma physics, nonlinear optics, general relativity, etc.) where they often serve as simplified effective models. In order for these models to be useful for engineering purposes as well, one has to understand the effect of inhomogeneities (usually modeled by a non-Euclidean domain geometry or a potential term) in making connections with the real world. Such a study would help set the parameters and limitations of not only the validity of the existing simplified models, but also the faithfulness of computer simulations (based on those models) in predicting experimental phenomena. On a more theoretical note, starting a mathematically rigorous study of the all-important physical theory of weak turbulence, as suggested in the proposal, represents a cross-disciplinary collaboration between pure mathematics, applied mathematics, and physics, with rewarding applications in plasma and fluid engineering, as well as ocean and atmospheric science.