Applied Analysis for Integrable Nonlinear Waves

自然界中的波(例如，水波或电磁波)可以通过某些微分方程式的解来模拟，这些微分方程式将各种物理过程合并到一个数学框架中，允许在原则上进行进一步的详细研究。但即使有了这样的波动方程，从模型中推导出重要信息仍然是一项艰巨的任务，这些信息是解决最初推动波浪研究的重要工程问题所需的信息。一种常见的方法是使用计算机(近似)求解方程。然而，这种方法的范围仅限于涉及特定初始条件的非常具体的模拟，并且仅在数值方法可以精确的参数范围内适用。另一方面，还有一些很常见的参数制度(例如，电磁波可以用光线近似的情况)，在这种情况下，基于计算机的方法变得困难，因此人们希望有一种替代的分析方法。这个项目的目的是为波传播问题开发这种渐近分析的替代方法，这些问题是非线性的(以便可以准确地模拟大浪)，但允许将它们与线性问题(适用于熟悉的叠加原理)相联系的一种变换。这种理论分析的一个应用将是描述由像深水地平线灾难这样的石油泄漏引起的地下石油羽流的演变。从渐近分析的角度(即进一步发展经典的定相法和积分最陡下降法)，本项目试图将完全可积的非线性波动方程置于与常系数线性方程类似的基础上。要解决的具体问题包括：小色散极限下Benjamin-Ono方程(重力内波的非局部可积模型)的逆散射变换的研究；半经典极限下波包通过二次非线性的共振相互作用的研究；半经典极限下描述二维空间波的散焦非线性薛定谔方程的类比研究；所谓流氓波的动力学稳定性的研究；半经典极限下可积方程的混合初边值问题的研究；以及“普遍波型”的研究。类似于统计力学和数学物理中普遍相变的波传播问题。

Waves in nature (for example, water waves or electromagnetic waves) can be modeled by solutions of certain differential equations that incorporate various physical processes into a mathematical framework that admits further detailed study in principle. But even with such a wave equation in hand, there remains the difficult task of deducing important information from the model, information that is needed to solve important problems of engineering that motivate the study of waves in the first place. One common approach is to use computers to solve the equations (approximately). However, such an approach is limited in scope to very concrete simulations involving particular initial conditions, and it holds only in parameter ranges in which the numerical methods can be accurate. On the other hand, there are other parameter regimes that are quite common (for example, the situation that electromagnetic waves can be approximated by light rays) in which the computer-based approach becomes difficult, and therefore one would like to have an alternative method of analysis. This project is aimed at developing such alternative methods of asymptotic analysis for wave propagation problems that are nonlinear (so that large waves can be accurately modeled) but that nonetheless admit a kind of transform relating them to linear problems (for which the familiar superposition principle applies). One application of such theoretical analysis would be to describe the evolution of sub-surface oil plumes caused by an oil leak like the Deepwater Horizon disaster. This project is an attempt to place completely integrable nonlinear wave equations on a similar footing as constant-coefficient linear equations, from the point of view of asymptotic analysis (i.e., to further develop nonlinear analogues of the classical methods of stationary phase and steepest descent for integrals). The specific problems to be addressed include the study of the inverse-scattering transform for the Benjamin-Ono equation (a nonlocal integrable model for internal gravity waves) in the small-dispersion limit, the study of the resonant interaction of wave packets through a quadratic nonlinearity in the semiclassical limit, the study of an analogue of the defocusing nonlinear Schroedinger equation describing waves in two spatial dimensions in the semiclassical limit, the study of dynamical stability of so-called rogue waves, the study of mixed initial/boundary-value problems for integrable equations in the semiclassical limit, and the investigation of "universal wave patterns", analogues to wave propagation problems of universal phase transitions in statistical mechanics and mathematical physics.