Inverse scattering transform outside of classical conditions

University Of Alaska Fairbanks Campus, Fairbanks AK

Investigators

Abstract

The investigator will study nonlinear partial differential equations that are used in applications, to model phenomena in hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, and astrophysics. The specific focus is on the so-called completely integrable systems, an area of applied mathematics also known as soliton theory. Much of what is currently known concerns rapidly decaying or periodic initial conditions, but physics and applications call for more general non-standard situations: the thrust of this project is on understanding the effect of slower decay (or no decay) initial data. The project will improve the modeling and predicting capabilities for rogue waves, integrable turbulence, propagation of coherent structures in noisy media, tidal waves, and meteorological phenomena (i.e., morning glory). The project will also offer research and mentoring opportunities at both undergraduate and graduate levels. Recruitment efforts will be mindful to broaden research participation in mathematical science, as well as target students intending to become high school mathematics teachers. The project studies in the context of the Korteweg-de Vries equation the effect that initial data with slower decay (or no decay) at spatial plus infinity have on the solutions of completely integrable systems, that is partial differential equations that can be solved and analyzed by means of the inverse scattering transform (IST). Approaching the case of slower decay at plus infinity presents severe mathematical challenges in the applications of the IST, and its study is a major unsolved open question. Predicting long-time behavior of the solutions is difficult, as the well-known powerful Riemann-Hilbert machinery breaks down on such initial profiles. The main effort of this project will be devoted to extending the Riemann-Hilbert method outside of the realm of classical situations. The investigator plans to use the methods of Hankel operators to tackle the arising issues and expects to discover new types of solutions, with more complicated wave structure, in contrast to the simplicity of existing solitons and radiation, or periodic (quasi-periodic) wave trains and their modulations. The results will have implications for applications and be relevant to the theory of the Schrodinger operator, the cornerstone of quantum mechanics, and the theory of Hankel and Toeplitz operators, fundamental objects of operator theory. This project is jointly funded by the DMS Applied Mathematics Program and the Established Program to Stimulate Competitive Research (EPSCoR). 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.