Nonlinear dispersive equations model physical phenomena arising in quantum mechanics, plasma physics, nonlinear optics, oceanography, all the way to general relativity. The purpose of this project is to improve the mathematical and scientific understanding of those equations via A) concrete research projects aimed at studying the long-time behavior of solutions to such equations, and B) an educational component aimed at introducing such problems to a new and diverse generation of mathematicians. The main focus is on the study of the so-called "non-equilibrium behavior" of nonlinear dispersive equations. Such a behavior is exhibited in many physically important phenomena, like turbulence in ocean waves or in the transmission of optical signals, to mention only a couple. While this turbulence is easy to observe and has very strong manifestations in nature, its scientific understanding is rather poor and the mathematical justification of the involved phenomena based on the model equations is rather difficult and is mostly still open. This project is aimed at rigorously proving turbulence phenomenon for certain nonlinear dispersive models.
The most important feature of the long-time behavior of nonlinear dispersive partial differential equations (PDE) on compact domains is out-of-equilibrium behavior or long-time instability. This means that solutions do not exhibit long-time stability near equilibriums solutions or configurations. Such questions can be addressed from a dynamical systems perspective and from a statistical physics perspective, in what is often called in the physics literature as "wave turbulence theory". The proposed projects address this broad problematic from both perspectives. In terms of dynamics, the aim is to construct solutions exhibiting the so-called "energy cascade" phenomenon, in which the energy of a dispersive system moves its concentration zone between characteristically different length scales. On the statistical physics perspective, projects aimed at justifying the fundamental equations of wave turbulence are proposed. These are effective equations for the dynamics that are derived in the physics literature from heuristic considerations, and have very important implications on the long-time behavior of dispersive PDE, provided they are rigorously justified. The project also includes an educational component aimed at raising the interest of a younger generation of researchers in those fundamental problems through workshops, seminars, REU projects, etc.
