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的解来建模，这些微分方程将各种物理过程合并到一个数学框架中，原则上允许进一步详细研究。 但是，即使有了这样一个波动方程，仍然存在从模型中推导重要信息的艰巨任务，这些信息是解决重要工程问题所需的，这些问题首先激发了对波动的研究。 一种常见的方法是使用计算机来求解方程（近似）。然而，这种方法的范围是有限的，非常具体的模拟，涉及特定的初始条件，它只适用于参数范围内的数值方法可以是准确的。 另一方面，还有其他相当常见的参数状态（例如，电磁波可以近似为光线的情况），其中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。 该项目旨在为非线性波传播问题开发渐近分析的替代方法（以便可以精确地模拟大型波），但仍然允许将其与线性问题（熟悉的叠加原理适用于线性问题）相关联的一种变换。 这种理论分析的一个应用是描述由深水地平线灾难等石油泄漏引起的地下石油羽流的演变。 这个项目是试图把完全可积的非线性波动方程放在与常系数线性方程类似的基础上，从渐近分析的角度来看（即，以进一步发展积分的定相和最速下降的经典方法的非线性类似物）。 具体问题包括Benjamin-Ono方程的逆散射变换研究（重力内波的非局部可积模型），通过半经典极限中的二次非线性研究波包的共振相互作用，在半经典极限下，对描述二维空间波动的散焦非线性薛定谔方程的模拟进行研究，研究所谓的流氓波的动力学稳定性，研究半经典极限中可积方程的混合初始/边界值问题，以及研究“通用波模式”，类似于统计力学和数学物理中通用相变的波传播问题。

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.