Nonlinear wave equations give a mathematical description for many phenomena in optics, fluid dynamics, and a variety of other physical systems. This research is aimed at the mathematical study of special and very important "solitary-wave" solutions that represent a single wave or a train of single waves traveling through the medium. Solutions of this mathematical nature are instrumental for predicting behavior and designing engineering devices for the wide range of physically unrelated phenomena, such as water wave dynamics, in particular, gigantic ocean waves ("rogue" waves), magnetization of materials, propagation of light in optical fibers and other optical media, or some quantum mechanical dynamics. Even when it is not possible to completely determine the solutions to such equations, the evolution of the corresponding physical system can often be understood satisfactorily by considering a computational or approximate solution. This is feasible, provided the behavior of solutions does not change qualitatively when unavoidable computational errors or uncertainty in the parameters are introduced. In mathematics this insensitivity to small perturbations is termed "stability." The principal goal of this research project is developing new mathematical tools for study nonlinear dispersive waves and stability of their solitary-wave solutions. In particular, the Principal Investigator (PI) aims at obtaining precise, quantitative information about the long time (asymptotic) behavior. Graduate students will be trained and mentored through their participation in this project.
The PI will consider the questions of stability of solitons, such as traveling waves, standing waves, traveling kinks, for various models arising in physical applications. Of particular concern will be the close-to-soliton behavior of the Dirac equations, the Ostrovsky and the short pulse equations, as well as various water wave equations. In addition, the PI will address several outstanding problems in the theory of spatially discrete dispersive systems, of the type of the discrete nonlinear Schroedinger equation and the granular chain model with Hertzian interactions. These are the models, where new paradigms are expected to emerge. More precisely, the models present obstacles to their investigation that are either not present in the corresponding "standard" continuous limits or else, the solutions behave in a substantially different ways than the respective continuous analogues. Thus, new mathematical methods need to be developed to address the challenges associated with the study of evolution of these discrete models.
