Stratified Fluids and Completely Integrable PDEs

在研究海洋中的水波和空气中的大气波时，习惯上使用渐近模型来代替一般的流体力学方程。这些模型通过抑制流体运动的不重要细节，提供了对潜在动力学的有效描述。这是通过对波幅、波长和流体深度施加一定的缩放制度来实现的。本项目关注的是对理解两层流体问题的一些主要渐近模型的长期行为至关重要的方法的发展。这种情景出现在由不同盐度的水分层形成的内部海浪中，例如对于包括厄尔尼诺效应在内的中期气候模拟以及长期气候预测都很重要。次表面内波也可以与表面波相互作用，产生像流氓波这样的现象，可以损坏或摧毁海洋船只或固定结构。为了从数学上研究潜在的方程，20世纪80年代提出了特殊解方法。这些方法试图将原方程中复杂的非线性动力学转化为可预测的线性动力学。然而，缺乏对这些方法可行性的严格数学分析。通过开展研究和教育活动的结合，PI将为这些方法开发数学理论，并与合作学科建立联系，同时吸引未来的学生进入相关领域。在当前的项目中，PI将开发Benjamin-Ono （BO）方程和中长波（ILW）方程的完全可积性理论。它们都是两层流体问题的一维渐近模型。BO和ILW是具有哈密顿结构的重要非线性色散方程。此外，我们还推测了被称为直接散射变换（DST）的形式作用角微分同态，将相空间上的动力学映射到无限维环面上的角平移。如果DST可以被反转（IST），人们将有一个强有力的方法来研究这些方程的长期渐近性。然而，目前没有针对BO的大数据IST理论，也没有针对ILW的小数据DST理论。与许多其他完全可积方程一样，BO和ILW的IST理论可以表述为Riemann-Hilbert （RH）问题。然而，BO和ILW的RH问题涉及某些非局部跳跃条件。非局部RH问题理论的缺乏阻碍了BO和ILW问题的解决。对于BO， PI将使用新发现的散射函数之间的关系将RH问题简化为零指标的Fredholm算子的逆。然后PI将尝试使用一个新的标识来证明这个Fredholm算子具有平凡的内核。对于ILW， PI将把DST问题重新定义为卷积型算子的反演，并证明卷积核上的某些一致估计。PI将通过正则化和在一定加权空间中的求解来构建ILW的IST理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the study of water waves in the ocean and atmospheric waves in the air, it is customary to use asymptotic models to replace general equations of fluid mechanics. These models provide efficient descriptions of the underlying dynamics by suppressing unimportant details of the fluid motion. This is achieved by imposing certain scaling regimes of the wave amplitudes, wavelengths, and fluid depths. This project is concerned with the development of methods critical in the understanding of long-time behavior of some major asymptotic models of the two-layer fluid problem. Such scenarios arise in internal ocean waves layered by water of different salinity, such as are important for medium-term climate modeling including the El Nino effect, as well as long-term climate prediction. Sub-surface internal waves can also interact with surface waves and produce phenomena like rogue waves that can damage or destroy ocean vessels or fixed structures. To study the underlying equations mathematically, special solution methods were proposed in the 1980s. These methods attempt to transform the complicated nonlinear dynamics in the original equations into well predictable linear dynamics. However, rigorous mathematical analysis of the feasibility of these methods is missing. By carrying out a combination of research and educational activities, the PI will develop the mathematical theory for these methods, and make connections to partner disciplines, while attracting prospective students into the related fields. In the current project, the PI will develop complete integrability theories for the Benjamin-Ono (BO) equation and the intermediate long wave (ILW) equation. They are both 1D asymptotic models for the two-layer fluid problem. BO and ILW are important nonlinear dispersive equations with Hamiltonian structures. Furthermore, formal action-angle diffeomorphisms known as the direct scattering transforms (DST) were conjectured to map the dynamics on phase space to angular translations on infinite dimensional tori. If the DST can be inverted (IST), one will have a powerful method to study long-time asymptotics of these equations. However, currently there is no large data IST theory for BO, nor is there even a small data DST theory for ILW. Like those for many other completely integrable equations, the IST theories for BO and ILW can be formulated as Riemann-Hilbert (RH) problems. Nevertheless, the RH problems for BO and ILW involve certain nonlocal jump conditions. Lack of theory for nonlocal RH problems hinders the solution of BO and ILW. For BO, the PI will use a newly discovered relation between the scattering functions to reduce the RH problem to inverting a Fredholm operator of zero index. The PI will then attempt to use a new identity to show that this Fredholm operator has trivial kernel. For ILW, the PI will recast the DST problem as inverting a convolution type operator and prove certain uniform estimates on the convolution kernel. The PI will construct the IST theory for ILW by regularization and solution in certain weighted spaces.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.