Asymptotic Regimes of Nonlinear Dispersive Equations

该项目的主要目的是探索影响非线性色散偏微分方程长时间行为的重要因素，如非线性薛定谔(NLS)、非线性波(NLW)、非线性Klein-Gordon(NKG)和Zakharov系统。首先，根据基本流形的体积增长和线性解的衰减之间的关系来研究区域几何的影响。然后在某些可处理的几何中，即欧几里德空间的商中，探讨了这种关系对非线性问题的含义。其次，考虑了一些弱非线性色散方程的小数据散射(渐近线性行为)问题。这些方程的非线性程度还不够高，经典方法不能成功。一个焦点是由势项表示的不均匀，以及这如何影响渐近行为。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域。在这一背景下，根据弱湍流理论的精神，采用适当的大盒子极限，导出了离散傅立叶模包络的新的连续介质方程。这个方程具有显著的对称性，甚至是显式解，并且可以(通过严格的近似结果)用来更好地理解原始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地产生于几个物理领域(等离子体物理、非线性光学、广义相对论等)。在那里，它们经常被用作简化的有效模型。为了使这些模型同样适用于工程目的，人们必须了解非均质性(通常由非欧几里德区域几何或势项建模)在与真实世界建立联系时的效果。这样的研究不仅有助于确定现有简化模型的有效性，而且有助于确定计算机模拟(基于这些模型)在预测实验现象方面的真实性。在更多的理论方面，按照提案中的建议，开始对至关重要的弱湍流物理理论进行严格的数学研究，代表着纯数学、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具有有益的应用。

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.