Soliton Dynamics and Scattering Theory

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The research is focused on understanding the mathematical aspects of soliton dynamics and related topics in wave propagation and scattering. The PI considers the nonlinear Schroedinger equation on one two and three dimensions. For attractive nonlinearities localized in space, solutions may exist, which can then move without dispersion, until they are perturbed. These class of equations play now a critical role in describing optical devices, Bose-Einstein condensate fluids and other nonlinear dispersive problems in fluid dynamics and plasma physics. The equations being nonlinear, require new techniques to understand the large time behavior of the solutions of this class of equations. Using the hydrodynamic formulation of Schroedinger equation it was possible to analyze in detail a completely new type of situations, in which a soliton solution is created through tunneling from a potential well. The analysis of the two dimensional case, with vorex initial condition, revealed a new phenomena, the formation and the ejection of waves in a form of jets. The theory we use, predict the direction and structure the of the jets. The prediction and the details of this soliton formation will be further studied. In particular, the method offers a new way of proving a-priori estimates on the solutions of nonlinear Schrodinger equation which are not of the standard energy estimates. A second part of the research involves the detailed decay in time of solutions of the wave equation on Schwarzschild and Kerr manifolds. In particular the decay rates as a function of the angular momentum of the initial data is pursued. This will provide a rigorous proof of a classical conjecture of Price in the theory of General Relativity.New processes involving optical devices are studied in detail. In particular one considers a situation in which light energy is located in a potential well, inside a properly active dielectric material. Then, the development of the localized energy is derived by a new mathematical formalism adapted to this situation. It is shown that soliton waves can emerge from the well, and the theory is capable of determining their size and speed. As such, it allows, by tuning the shape of the potential well and the incoming energy to produce solitons with desired profile, sometimes referred to as soliton guns. This approach has already been observed in some experiments based on previous works by the PI. the recent preditin of jet formation through tunneling two dimensional systems, is being developed; in particular, experimental observation and validation is planned. A second part of the research is the mathematical analysis of wave propagation and decay on manifolds generated by black holes. The approach to this problem led a to a new, more general theory of solving integral equations of the Voltera type, and also led to a rigorous mathematical verification of some classical conjectures in the physics literature concerning the behavior of radiated waves off a black hole.