New Directions in the Asymptotics of Nonlinear Waves

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。这样的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，例如长时间或小色散，通常也可以令人满意地理解系统。对于有些理想的可积波动方程，给出了一个丰富而成熟的渐近理论。然而，为了对更现实的物理系统提供准确的预测，有必要扩展当前的模型，以考虑包括耦合或干扰、高阶修正和边界在内的问题。通过利用最新的数学进展，该项目将改进现有的方法，以更好地模拟这三种效应。所研究的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湍流。本研究旨在通过以下三个项目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系统，包括三波共振相互作用方程，建立更好的耦合系统数学模型。(2)建立近似可积方程（如正弦-戈登方程的哈密顿摄动）的长期行为和不稳定性开始的结果，以便更好地纳入可能不可忽略的高阶物理效应。(3)扩展了统一变换方法来理解有限或半有限域上的小色散行为。对离焦非线性薛定谔、大质量Thirring和相关方程的分析将改进边界效应重要的模型。

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.