Stratified Fluids and Completely Integrable PDEs

University Of Oklahoma Norman Campus, Norman OK

Investigators

Abstract

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.