Monday, March 20 at 3:30pm to 4:30pm
Kaprielian Hall (KAP), 414
3620 South Vermont Avenue, Los Angeles, CA 90089
Zaher Hani, Georgia Tech
In this talk, 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 time scales (and their corresponding effective dynamics) that hold for more-or-less generic dispersive equations, we will try to go further in describing the effective dynamics over much longer time scales. This becomes more equation-dependent, and here we specialize to the nonlinear Schroedinger equation (any power nonlinearity) posed on a large box of size $L$. Our main result is to exhibit a new type of dynamics that appears at a particular large time scale, (that we call the resonant time scale) defined in terms of the size of the domain L and the characteristic size of the initial data. As mentioned, going to such long time scales is partly motivated by turbulence theory for dispersive PDE, aka wave turbulence theory, in which one would like to address the effective dynamics on even longer timescales. We will touch on these topics and time scales as well. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah (all at Courant Institute, NYU).