Integrable PDEs and Hankel operators

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This project is devoted to some fundamental issues of propagation of waves in various media. The principal methods of this work come from the soliton theory. Solitons are very special solitary waves ("bumps") of water that move with constant speed without any deterioration in their shape. The first soliton siting was described in 1834 by the Scottish naval architect John Scott Russell, who noticed this wave in a channel and pursued it on his horse for quite a while. The soliton theory was originated in the mid-1960s from the fundamental Gardner-Greene-Kruskal-Miura discovery of the inverse scattering transform (IST) for the Korteweg-de Vries (KdV) equation for shallow water waves (this equation happens to describe the channel phenomenon that Russell observed). Soon thereafter different versions of IST were found for many other physically important nonlinear evolution partial differential equations (PDEs) referred to as completely integrable systems. Being conceptually similar to the Fourier transform, the IST has yielded a tremendous amount of information about completely integrable systems, far beyond what standard PDE techniques may offer. Soliton theory is regarded as a major achievement of the 20th century science connecting different branches of pure mathematics and theoretical physics with numerous applications ranging from hydrodynamics and nonlinear optics to astrophysics and elementary particle theory. Much of work in soliton theory has been done on the propagation of waves initiated by rapidly decaying or periodic initial data (the so-called classical data). 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 classical data meets principal difficulties that are yet to be surmounted. The project will focus on soliton theory for initial profiles that are much broader than classical. We expect new types of solutions with much more complicated wave structure and far-reaching practical applications. It is expected that the results could be used for understanding rogue waves, soliton propagation on different backgrounds (including noisy), tidal waves, certain meteorological phenomena (e.g. morning glory), or for the study of propagation of coherent structures in noisy media in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, astrophysics, etc. In the context of the KdV equation the principal investigator has reformulated the classical IST in terms of Hankel operators and Titchmarsh-Weyl m-functions that let one extend the IST to a surprisingly broad class of initial data. This was achieved by employing some subtle properties of the m-function and deep results from the theory of Hankel operators. The principal investigator plans to use powerful methods of Hankel operators to identify the broadest possible class of initial profiles for which a suitable analog of the IST exists. Another objective is asymptotic analysis of the underlying solutions. The well-known powerful machinery of the classical Riemann-Hilbert problem breaks down on such initial profiles in a number of serious ways. The main thrust will be put on understanding how to make the Riemann-Hilbert problem work far outside of the realm of classical problems. The results are expected to be instrumental for various applications. The accompanying mathematical problems are also very important 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. Uncovering connections between soliton theory and Hankel operators theory is of great independent interest and could potentially have a profound influence on both theories. The project will have a very large educational component. The principal investigator is committed to continuing his research experience for undergraduates program on nonlinear wave phenomena 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.