Monday, October 14 at 3:30pm to 4:30pm
Kaprielian Hall (KAP), 414
3620 South Vermont Avenue, Los Angeles, CA 90089
Zaher Hani, University of Michigan
Abstract: Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the "wave kinetic equation” (WKE). This is the wave-analog of Boltzmann's equation for particle collisions. A fundamental scientific question to resolve here is to provide a rigorous derivation of this kinetic equation, in a way that allows to justify its significance in describing the long-time dynamics of the Hamiltonian dispersive PDE we started with. In this talk, we shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and provide a rigorous derivation of its kinetic equation on timescales that are significantly shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah).