On the Long Time Behavior of Nonlinear Dispersive Equations

Waves are ubiquitous in nature and described by so-called dispersive equations, from the light transmission in fiber optics to the charge transport between base pairs in the DNA molecule to the particle interaction inside atoms. These models are pervasive in physical and biological phenomena, but their long-time behavior is far from well-understood. Here long-time behavior is a general term used to encompass the notions of global well-posedness, scattering effects, unique continuation of the solutions, and wave turbulence. Understanding the evolutionary behavior of dispersive equations has a broad range of applications in fields, such as, numerical simulations, optics, condensed matter, fluid mechanics, and biology. The aim of this project is to understand the long-time dynamics of dispersive systems while bringing new insights that will establish novel approaches and models in applications. The project includes research training opportunities for undergraduate and graduate students. This project contains three components. The first component concerns the unique continuation principle, which is believed to play an important part from a numerical point of view. In numerical simulations, a computer often preforms computations on finite approximations of the domain or the data or both, which may result in non-uniqueness or instability issues. A well-developed unique continuation theory will provide better stability and reliability of the computational results. The second considers dispersive models on waveguides, an important model for data transmission. The investigator will also introduce a model with waveguide behaviors that for the first time will connect a probabilistic well-known Ornstein-Uhlenbeck operator to dispersive equations and has the potential to build a strong bridge between dispersive partial differential equations and probability. The last component investigates the energy transfer phenomenon, which relates to the wave turbulence theory. Wave turbulence theory has the feature of universally predicting the evolution of the wave action spectral density of interacting wave systems, which will help with forecasting surface gravity waves in the oceans, among others. 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.