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.

该项目在通用波方程的散射理论领域。目的是表征数学物理学中复杂型非线性方程的解决方案的大型行为，这些解决方案描述了量子系统和非线性光学模型中的波传播等。找到这种系统的可能不对称状态对于对物理现象和应用的定性和定量理解至关重要。例如，在量子力学中找到被激光脉冲扰动的分子的分解成分的基本问题是这种类型的。同样，光脉冲的能量损失在光纤中移动了很长的距离，而时间依赖噪声对量子和光学设备稳定性的影响就是示例。该项目的目的是为一般类别的非线性schrödinger方程寻找解决方案的不对称行为和其他特性。该项目的关键部分是找到所有可能的不对称状态。对于非线性方程来说，这是一项非常困难的任务，一般性质的结果很少。方程式被认为是必要的分析，计算和数值工具的复杂性。将开发现代功能分析方法和物理见解的结合。在球形对称的初始数据和扰动项的情况下，这也可能取决于时间和空间变量，目标是证明该解决方案分为自由波和（弱）局部零件，并得出本地化部分的属性。该项目包括对具有多个Ergodic成分的动力学方程的分析和数值研究。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估被认为是宝贵的支持。