This project involves theoretical and applied investigations of certain ubiquitous nonlinear wave phenomena. More specifically, the systems that will be studied belong to a class of nonlinear evolution equations referred to as soliton equations or integrable systems, which arise as mathematical models in various areas of applications, ranging from fluid dynamics and nonlinear optics, to low-temperature physics and Bose-Einstein condensates, to name a few. In particular, one of the model equations that will be studied describes the formation of rogue waves in the open ocean. These extreme events, which are characterized by wave crests up to four times bigger than the average, have been known to cause significant damages to vessels and other equipment. Effects similar to rogue waves have also been recently observed in optical fibers. A precise mathematical description of these model equations is also a key component in the design of optical fiber transmission systems. The project outcomes will help to better characterize the behavior of these systems and the properties of their solutions, they will elucidate the role that such solutions play in the generation of rogue waves, and they will contribute to paving the way for a deeper understanding of these important nonlinear phenomena. The project will provide a rich educational experience through research for graduate students at the State University of New York at Buffalo and undergraduate students at the University of Colorado at Colorado Springs.
Mathematically, the overarching goal of the project is a wide-ranging investigation of initial and initial-boundary value problems for integrable nonlinear evolution equations, thereby contributing to closing the gap between our knowledge of linear and nonlinear integrable systems. More precisely, the project comprises problems where the boundary conditions play a key role, as well as various kinds of singular limits. Specific problems that will be studied include: (i) the development of the inverse scattering transform for coupled systems of equations of nonlinear Schrodinger (NLS) type with non-zero boundary conditions at infinity; (ii) the study of initial-boundary value problems for NLS equations on a finite interval with linearizable, periodic or nearly periodic boundary conditions; (iii) the study of the long-time asymptotics of NLS systems affected by modulational instability and of semiclassical and dispersionless limits for NLS equations and the Korteweg-deVries equation with periodic boundary conditions.
