Asymptotic Dynamics of Nonlinear Wave and Dispersive Equations

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Many wave propagation phenomena in the natural sciences and in engineering can be modeled by nonlinear wave and dispersive equations. While the theory of linear wave equations predicts that waves spread out and decay as time goes by, once nonlinear effects are taken into account, this changes drastically. In particular, linear and nonlinear effects may balance out to create so-called soliton solutions, whose shapes persist and refuse to disperse. It is widely believed that solutions to most nonlinear wave and dispersive equations with "generic" initial data should eventually decompose into a finite number of solitons plus a radiative term that goes to zero. This project concentrates on two themes that play an important role in the quest to understand this grand picture how waves propagate overtime. The principal investigator (PI) will develop new methods and techniques for the study of the asymptotic stability of solitons in the presence of strong nonlinear interactions. Asymptotic stability refers to the phenomenon that if a soliton gets pushed a little bit, it may wiggle for a while, but ultimately return to a form similar to the one it began with. Further, the PI will investigate the long-time dynamics of solutions to nonlinear wave equations with generic randomized initial data in several novel regimes.More specifically, the PI will use harmonic analysis and vector field techniques together with tools from spectral theory and probability theory to work toward the following goals: (1) carry out a program to obtain precise asymptotics of small solutions to one-dimensional Klein-Gordon equations with variable coefficient nonlinearities, which are related to asymptotic stability questions for "kink" solitons in numerous field theories in physics; (2) develop a modulational approach for proving the (co-dimensional) stability of certain solitons arising in some quasilinear geometric wave equations; and (3) initiate the study of the long-time dynamics of solutions to geometric wave equations and to wave equations with long-range nonlinearities for random initial data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.