In this minicourse, we will be mainly concerned with the following question: Suppose we consider a nonlinear dispersive or wave equation on a large domain of characteristic size L: What is the effective dynamics when L is very large? This question is relevant for equations that are naturally posed on large domains (like water waves on an ocean), and in turbulence theories for dispersive equations. It’s not hard to see that the answer is intimately related to the particular time scales at which we study the equation, and one often obtains different effective dynamics on different timescales. After discussing some relatively “trivial” time scales (and their corresponding effective dynamics), we shall attempt to go to longer times scales and try to describe the effective equations that govern the dynamics there. The ultimate goal is to reach the so-called the “kinetic time scale” over which it is conjectured that the effective dynamics are described by a kinetic equation called the “wave kinetic equation”. This is the main claim of wave turbulence theory. We will discuss several recent works, obtained in collaboration with Tristan Buckmaster, Erwan Faou, Pierre Germain, and Jalal Shatah , that are aimed at addressing the above problematic for the nonlinear Schrodinger equation.
Location:
- A-132 on Tuesday 12/06
- A-005 on Thursday 14/06
- A-133 otherwise