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New Directions in the Asymptotics of Nonlinear Waves

非线性波渐进的新方向

基本信息

批准号：
1615718
负责人：
Robert Buckingham
金额：
$ 20.89万
依托单位：
University of Cincinnati Main Campus
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1615718&HistoricalAwards=false
关键词：
New Directions Asymptotics Nonlinear Waves

项目摘要

Nonlinear wave equations give a mathematical description for many phenomena in optics, fluid dynamics, and a variety of other physical systems. Such a description is instrumental for predicting behavior and designing engineering devices, such as optical switches for information transmission. Even when it is not possible to completely determine the solutions to such equations, the system can often be understood satisfactorily by considering an appropriate approximation, such as long time or small dispersion. A rich and maturing asymptotic theory has been developed for the somewhat idealized integrable wave equations. However, in order to provide accurate predictions for more realistic physical systems, it is necessary to extend the current models to account for issues including coupling or interference, higher-order corrections, and boundaries. By taking advantage of recent mathematical advances, this project will improve available methods to better model these three effects. Specific applications of the models studied include the development of all-optical switches, Raman scattering, flux propagation in superconducting Josephson junctions, and hydrodynamic turbulence. The research aims to enhance the physical applicability of current models of nonlinear wave propagation through the following three projects: (1) Extension of the small-dispersion theory to multicomponent systems, including the three-wave resonant interaction equations, to develop better mathematical models of coupled systems. (2) Establishing results on the long-time behavior and onset of instabilities in near-integrable equations, such as Hamiltonian perturbations of the sine-Gordon equation, in order to better incorporate higher-order physical effects that may not be negligible. (3) Extending the unified transform method to understand small-dispersion behavior on finite or semi-finite domains. Analysis of the defocusing nonlinear Schrodinger, massive Thirring, and related equations will improve models where boundary effects are important.

非线性波动方程为光学、流体动力学和其他各种物理系统中的许多现象提供了数学描述。这种描述有助于预测行为和设计工程设备，如用于信息传输的光开关。即使不可能完全确定这些方程的解，通过考虑适当的近似，如长时间或小色散，通常也可以令人满意地理解系统。对于理想化的可积波动方程，已经发展出了丰富而成熟的渐近理论。然而，为了提供更真实的物理系统的准确预测，有必要扩展当前的模型，以解决耦合或干扰，高阶校正和边界等问题。通过利用最新的数学进步，该项目将改进现有的方法，以更好地模拟这三种影响。研究模型的具体应用包括全光开关的发展，拉曼散射，超导约瑟夫森结中的通量传播，和流体动力学湍流。本研究旨在通过以下三个方面的工作来提高现有非线性波传播模型的物理适用性：（1）将小色散理论推广到多组分体系，包括三波共振相互作用方程，以建立更好的耦合体系数学模型。(2)建立关于近可积方程的长时间行为和不稳定性开始的结果，例如正弦-戈登方程的哈密顿扰动，以便更好地纳入可能不可忽略的高阶物理效应。(3)扩展统一变换方法以理解有限或半有限域上的小色散行为。分析散焦非线性薛定谔，大量的Thirring，和相关的方程将改善模型的边界效应是重要的。