Integrable PDEs beyond standard assumptions on initial data

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This project is devoted to the study of some fundamental problems of soliton theory. A soliton is a special type of wave that shows a remarkable stability when traveling through various media. Examples include such well-known phenomena as tsunami waves and pulses in optical fibers. The first observation and scientific description of a soliton was given by Scott Russell in 1834. The equation describing what Russel had observed was derived in 1895 by Korteweg and de Vries but it was not until 1967 when this equation, now called Korteweg-de Vries (KdV), was solved in closed form by Gardner, Greene, Kruskal, and Miura. Their method is regarded as a major achievement of the 20th century science. It gave rise to soliton theory, applicable to broad classes of physically important evolution partial differential equations, ranging from hydrodynamics of water waves (rogue waves in the ocean) and nonlinear optics (propagation of information in optical fibers) to astrophysics, atmospheric sciences, and elementary particle theory. This project will develop novel approaches to extend the theory to the physically and practically important cases of slowly decaying waves, which are still beyond the reach of the current methods. The project will have a very large educational component. The investigator is committed to continuing his research experience for undergraduates program on nonlinear wave phenomena. This program is designed to identify and mentor young scholars in the field of applied mathematics. It is his intent to attract a diverse (gender, ethnicity, disability) group of talented undergraduates into the program to broaden the participation of underrepresented in the mathematical sciences groups.Integrable systems have been primarily studied in the connection with propagation of waves initiated from rapidly decaying or periodic initial data. In the KdV context, the corresponding solutions have a relatively simple and well understood wave structure of running solitons accompanied by radiation of decaying waves, or periodic wave-trains and their modulations. However, any deviation from such data meets principal difficulties. The main thrust will be put on understanding of the effect of slower decay (or even no decay) at spatial plus infinity. Physical motivations include modeling rogue waves, nonlinear wave propagation in (pseudo) periodic media with slowly decaying amplitude, integrable turbulence, and propagation of coherent structures in noisy media. From the mathematical viewpoint, it is an uncharted territory. Slower decay at plus infinity causes serious complications at every step of the IST. The main effort will be put on understanding how to make the method of the Riemann-Hilbert problem, a modern powerful machinery of asymptotic analysis, work far outside of the realm of classical problems. To this end, developing direct/inverse scattering theory for long-range potentials will be required. The investigator expects to find new types of solutions with far-reaching practical applications, which include, but not limited to, the understanding of rogue waves, soliton propagation on different backgrounds, and the study of propagation of more general coherent structures in noisy media appearing in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, astrophysics, and other areas where integrable systems naturally arise.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.