Linear and Nonlinear Dispersive Waves: Solitons, Nonlinear Resonances and Spectral Theory

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The research in this project is focused on the analysis of wave dynamics. Wave propagation and other properties are the backbone of modern Science and Technology. Quantum waves control the nano world, electromagnetic waves manifest as light, lasers, heat, and are responsible for chemical reactions and electronic devices. Gravity theory is described by Einstein wave equation as well. The analysis of complex systems via such wave equations is extremely difficult, and even super fast computers cannot do the job. Therefore, deeper understanding of the equations provides new tools for the relevant features of the solutions. At the same time, the acquired knowledge opens new directions of research in mathematics. In this proposal the behavior of special solutions, called coherent states, of fundamental equations of mathematical physics are studied. In particular, focus is given to the effect of disturbances of such solutions over long periods of time, in an effort to control the stability, life time and evolution of such solutions. The cases mostly considered are solutions called solitons- which are clumps of self trapped waves in some small domain. The solitons formed by laser beams going through an optical fiber play a key role in fast communication and other future planned optical devices.Soliton dynamics of the nonlinear Schroedinger equation will be studied. While the complete understanding of the solutions of such equations for all initial data seems remote, a novel direction is proposed: The interaction of solitons with radiation, with large potential terms, and with each other, will be analyzed by identifying processes which are adiabatic in time. Relevant adiabatic dispersive theory will be used, previously developed by the PI and new planned methods. It is expected to complete a gap in our understanding of a fundamental aspect of nonlinear scattering: interaction process of a soliton with large perturbation. The study of nonhomogeneous nonlinearities of the long range type, initiated by the PI, which are fundamental to many nonlinear scattering problems (e.g. Kink scattering), uncovered a new, subtle resonance phenomena: the unbound growth of some Invariant Sobolev norm of such systems. It is the aim of the research to understand the implications of these new processes. Dispersive estimates for linear equations play a central role in spectral and scattering theory. The PI's recent and future work is to develop an alternative, abstract theory. By avoiding the need to study the explicit (eigen) solutions or fundamental solutions of the linear equation, one can expand the dispersive theory to new classes of problems, including dynamics on manifolds.