The PI will apply the techniques of modern analysis to some outstanding open problems in the stability theory of breathers in the discrete Schr""odinger equation and Klein-Gordon equation context as well as special solutions for dispersive partial differential equations. The project concentrates on the basic questions concerning stability as well as the existence and the precise description of stable manifolds for solutions, with a few unstable directions. While the stable scenarios have received a great deal of attention in the last twenty years or so, the behavior close to unstable solutions has been less well-studied. One reason is that unstable structures present themselves in a more challenging environment in terms of the mathematical techniques that must be used. This is especially true in the presence of a marginally stable spectrum, resonant edges of the essential spectrum and in the low dimensional cases, all of which will be of primary interest in this project.
The project will focus on the study of nonlinear dispersive equations, which model mathematically important processes, such as propagation of light in optical medium. Some of these models arise in the study of quantum mechanical systems and nonlinear optics, while others find their roots in fluid dynamics. These problems can also present themselves as a control problem in stabilization theory. Roughly speaking, if one starts close to an unstable configuration, how does one make only small adjustments along the way, in order to stay close to the initial configuration? Better mathematical description of the behavior of the solutions of these equations, especially their asymptotic behavior in time and space, will greatly improve our understanding of the underlying physics and it will help in the development of technologies that use them.
In the modern theory of partial differential equations, special attention is paid to solutions with special properties, which are observed in experiments and real physical situations. There are various types of these, so called solitons or coherent structures or waves - pulses, fronts, backs, traveling waves, standing waves etc. each one arising in different models and of great practical significance. There were two overarching main goals of the project. The first goal was to show (in a mathematically rigorous way) the existence of such waves and supplement these studies with appropriate numerical simulations. The second goal was to study the stability of these structures, namely starting close to such solutions, does the configuration stay close to it (in appropriate sense) forever or does it diverge from this state. Clearly, both questions of existence and stability of such solutions are of fundamental importance to the understanding of the physics behind the underlying processes. The PI and collaborators have made significant progress in both themes. We only report on the main and notable achievements. In a series of papers, the PI and P. Kevrekidis have shown the existence of bell-shaped traveling waves for the problem of (infinite) monomer chains, interacting according to the Hertz law. The results also provided a rigorous justification of the double exponential rate of decay of these waves, coined ``compactons''. They have also rigorously justified a list of other useful properties, which were experimentally and numerically observed by other scientists. In another direction, in a series of papers, the PI and Stanislavova have fully characterized the spectral/linear stability of a class of solutions arising in second order in time Hamiltonian partial differential equations. These models are ubiquitous in the modeling of various processes in actual physical systems. This research extends our ability to decide (in many cases for all parameter values!) about the stability/instability of such waves in models, which were previously inaccessible through the currently used methodology. All findings of this work have been published in prestigious, peer-reviewed journals. In addition, the main scientific outcomes were reported at international conferences, departmental seminars and colloquiums. In the area of outreach/human resources, and with the support of the grant, the PI has trained and supported a Ph.D. student, who has since graduated and is currently employed as a postdoctoral fellow in a large state University. In addition, two undergraduate students have been trained, as part of an REU supplemental research, in the mathematical and numerical methods relevant to the project.
