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.

这个项目是在一般波动方程的散射理论领域。其目的是描述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。寻找这类系统可能的渐近态对于定性和定量地理解物理现象和应用都是至关重要的。例如，量子力学中的基本问题，即找到被激光脉冲扰动的分子的分裂分量，就是这种类型。类似地，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。本项目的目的是寻找一般类非线性薛定谔方程解的渐近行为和其他性质。该项目的一个关键部分是找到所有可能的渐近状态。这是一个非常困难的任务，非线性方程，一般性质的结果是稀缺的。所考虑的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。在球对称的初始数据和扰动项的情况下，这也可以取决于时间和空间变量，我们的目标是要显示的解决方案分为一个自由波和（弱）本地化部分，并获得本地化部分的属性。该项目包括对具有多个遍历分量的动力学方程的分析和数值研究。该奖项反映了NSF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。