Integrable Partial Differential Equations Beyond Standard Assumptions on Initial Data

University Of Alaska Fairbanks Campus, Fairbanks AK

Investigators

Abstract

This project is concerned with some fundamental problems of soliton theory which deals with nonlinear wave propagation in various media. Solitons are very special solitary waves that move with constant speed without changing their shape; the most prominent example is a tsunami wave. The first scientific description of a soliton was given in 1834 by Scott Russell. The equation describing what Russel had observed was derived in 1895 by Korteweg and de Vries (KdV), but it was not until 1967 when the KdV equation was solved in closed form. The method of solution, the inverse scattering transform (IST), is regarded as a major achievement of the 20th century mathematics. It gave rise to soliton theory that is dealing with broad classes of physically important differential equations which can be solved by a suitable IST (such equations are also called completely integrable systems). The range of applications is enormous: from hydrodynamics and nonlinear optics to astrophysics and elementary particle theory. Integrable systems have been primarily studied in the connection with propagation of waves initiated by rapidly decaying or periodic initial data (the "classical" data). The corresponding solutions have a relatively simple wave structure of traveling solitons accompanied by radiation of decaying waves, or periodic wave-trains and their modulations. However, any deviation from classical data leads to fundamental difficulties; it is the main focus of the project to overcome them. Entirely new types of solutions, with much more complicated wave structure and far-reaching practical applications, are expected to arise. The results could be used for understanding rogue waves, soliton propagation on different backgrounds (including noisy), tidal waves, certain meteorological phenomena (i.e. morning glory), or the study of propagation of coherent structures in noisy media (or in a general wave setting), in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma physics, astrophysics, etc. The project will have a very large educational component. The principal investigator (PI) will continue his research experience for undergraduates program to identify and mentor young scholars in the field of applied mathematics. In the KdV setting the PI has reformulated the IST in terms of Hankel operators and Weyl m-functions. It lets one extend the IST to a surprisingly broad class of initial data. The PI plans to continue using these powerful tools to identify the broadest possible class of initial data for which a suitable analog of the IST exists. Another objective is asymptotic analysis of the underlying solutions. The most powerful approach is based on the Riemann-Hilbert (RH) problem which also breaks down on such initial data in a number of serious ways. The main thrust will be put on understanding how to make the RH 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 Schrödinger operator, and the theory of Hankel and Toeplitz operators, fundamental objects of operator theory. Uncovering connections between soliton theory and Hankel operators is of great independent interest and could potentially have a profound influence on both theories.