Global Dynamics of Nonlinear Dispersive Evolution Equations and Spectral Theory

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This project aims at understanding the propagation of waves in a wide sense. On the one hand, the PI will investigate the behavior of waves in space, as they interact nonlinearly with themselves and matter over large space-time scales. The ultimate goal is to explain how the energy, which is stored in a wave undergoing a nonlinear dynamical evolution, ultimately splits into quantized pieces and a wave "at the horizon". The latter refers to energy, possibly of large size, which is infinitely spread out and does not interact with anything in a noticeable fashion. In contrast with this macroscopic behavior, the project also aims at understanding the behavior of waves on the microscopic scale, such as in crystals or quasi-crystals. The goal is to explain transitions from an insulating state to that of a conductor, which these materials may exhibit as they undergo changes on the molecular level. Such changes may occur through the insertion of impurities, or changes in the environment. Both the macroscopic as well as the microscopic behavior of waves is of crucial importance to science and engineering, and profoundly affects our daily modern lives. Modern communication relies on waves transmitted over large distances both in space but also along glass fiber cables. For the latter the properties of the material are crucial and both nonlinear effects as well as aforementioned microscopic phenomena decide the suitability of the underlying medium. More technically speaking, the PI intends to further investigate the rigorous mathematical theory of focusing dispersive semilinear evolution equations. A major open problem is to analyze the resolution of any solution into moving solitons and radiation. Some success has been achieved in recent years on this important problem, but for nonintegrable equations we are far from a satisfactory understanding. The PI is currently involved in the study of this problem in the dissipative setting in which some damping is added to the equation. The Hamiltonian setting appears to be very difficult at the moment, especially in the subcritical regime. The methods involved derive from dynamical systems, invariant manifold theory, and dispersive PDEs. The quantum mechanical problems alluded in the previous paragraph belong to the area of Anderson localization. Together with his long-standing collaborator Michael Goldstein at Toronto, but also with young collaborators which are joining the field, the PI intends to bring the body of techniques which were developed over the past 20 years based on large deviation estimates, the avalanche principle, semi-algebraic sets, and harmonic analysis such as (pluri)subharmonic functions and the Cartan estimate, to bear on both linear and nonlinear problems in dynamical systems and spectral theory. Ultimately, the goal here is also to better describe the behavior of wave propagation in disordered media.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.