The Asymptotic Solutions of Dispersive and Hyperbolic Equations

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This project is in the field of scattering theory of general wave equations. The aim is to characterize the large-time behavior of solutions of complex-type nonlinear equations in mathematical physics, which describe wave propagation in quantum systems and nonlinear optics models, among others. Finding the possible asymptotic states of such systems is critical to both qualitative and quantitative understanding of the physical phenomena and to applications. For example, the fundamental question in quantum mechanics of finding the breakup components of a molecule that is perturbed by a laser pulse is of this type. Similarly, the energy loss of light pulses moving a long distance in an optical fibre and the effects of time dependent noise on the stability of quantum and optical devices are examples.The aim of this project is to find the asymptotic behavior and other properties of the solutions for a general class of nonlinear Schrödinger equations. A key part of the project is to find all possible asymptotic states. This is an exceedingly difficult task for nonlinear equations, and results of general nature are scarce. The complexity of the equations considered necessitates different analytical, computational, and numerical tools. A combination of modern functional analytic methods and physical insights will be developed. In the case of spherical symmetric initial data and perturbation term, which can also depend on time and space variables, the goal is to show that the solutions break into a free wave and a (weakly) localized part and to derive the properties of the localized part. The project includes the analytic and numerical study of kinetic equations with multiple ergodic components.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.