The Asymptotic Solutions of Dispersive and Hyperbolic Equations

本课题属于一般波动方程的散射理论领域。目的是表征数学物理中复杂型非线性方程解的大时间行为，这些方程描述了量子系统和非线性光学模型中的波传播。找到这种系统可能的渐近状态对于物理现象的定性和定量理解以及应用都是至关重要的。例如，在量子力学中，寻找被激光脉冲扰动的分子的分解成分的基本问题就是这种类型的。同样，光脉冲在光纤中长距离移动的能量损失和时间依赖性噪声对量子和光器件稳定性的影响都是例子。本课题的目的是找出一类一般非线性Schrödinger方程解的渐近性质和其他性质。这个项目的一个关键部分是找到所有可能的渐近状态。对于非线性方程来说，这是一项极其困难的任务，而且一般性质的结果很少。考虑到方程的复杂性，需要不同的分析、计算和数值工具。将发展现代功能分析方法和物理见解的结合。在球对称初始数据和扰动项的情况下，它也可以依赖于时间和空间变量，目标是证明解分解成自由波和（弱）局部化部分，并推导出局部化部分的性质。该项目包括多遍历分量动力学方程的解析和数值研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is in the field of scattering theory of general wave equations. The aim is to characterize the large-time behavior of solutions of complex-type nonlinear equations in mathematical physics, which describe wave propagation in quantum systems and nonlinear optics models, among others. Finding the possible asymptotic states of such systems is critical to both qualitative and quantitative understanding of the physical phenomena and to applications. For example, the fundamental question in quantum mechanics of finding the breakup components of a molecule that is perturbed by a laser pulse is of this type. Similarly, the energy loss of light pulses moving a long distance in an optical fibre and the effects of time dependent noise on the stability of quantum and optical devices are examples.The aim of this project is to find the asymptotic behavior and other properties of the solutions for a general class of nonlinear Schrödinger equations. A key part of the project is to find all possible asymptotic states. This is an exceedingly difficult task for nonlinear equations, and results of general nature are scarce. The complexity of the equations considered necessitates different analytical, computational, and numerical tools. A combination of modern functional analytic methods and physical insights will be developed. In the case of spherical symmetric initial data and perturbation term, which can also depend on time and space variables, the goal is to show that the solutions break into a free wave and a (weakly) localized part and to derive the properties of the localized part. The project includes the analytic and numerical study of kinetic equations with multiple ergodic components.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.