The Asymptotic Solutions of Dispersive and Hyperbolic Equations

色散方程和双曲方程的渐近解

• 批准号: 2205931
• 负责人: Avraham Soffer
• 金额: $ 25万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2205931&HistoricalAwards=false
• 关键词: Asymptotic; Solutions; Dispersive; Hyperbolic; Equations

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.

该项目属于一般波动方程的散射理论领域，其目的是表征数学物理中复杂类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等中的波传播。找到此类系统可能的渐近态对于物理现象的定性和定量理解及其应用至关重要，例如，量子力学中寻找受激光脉冲扰动的分子的破裂成分的基本问题是。同样，光脉冲在光纤中长距离移动的能量损失以及时间相关噪声对量子和光学器件稳定性的影响都是例子。该项目的目的是找到渐近行为和一般非线性薛定谔方程组解的其他性质 该项目的一个关键部分是找到所有可能的渐近状态，这对于非线性方程和一般方程的结果来说是一项极其困难的任务。在球对称初始数据和扰动项的情况下，所考虑的方程的复杂性需要不同的分析、计算和数值工具。关于时间和空间变量，目标是证明解分解为自由波和（弱）局域部分，并推导局域部分的属性。该项目包括动力学的分析和数值研究。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。