On the wave turbulence theory for a stochastic KdV type equation

04/06/2022 1:30 pm - 2:30 pm
CMSA Room G10
Address: CMSA, 20 Garden Street, Cambridge, MA 02138 USA
Random Matrix & Probability Theory Seminar
Speaker: Minh-Binh TRAN (SMU & MIT)

Location: CMSA, Room G02

Title: On the wave turbulence theory for a stochastic KdV type equation

Abstract: We report recent progress, in collaboration with Gigliola Staffilani (MIT), on the problem of deriving kinetic equations from dispersive equations. To be more precise, starting from the stochastic  Zakharov-Kuznetsov equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a derivation of the 3-wave kinetic equation. We show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave  kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension is  $d\ge 2$, the smallness of the nonlinearity $\lambda$ is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation.  Unlike the cubic nonlinear Schrodinger equation, for which such a general result is commonly expected without the noise, the kinetic description of the deterministic lattice ZK equation is unlikely to happen. One of the key reasons is that the dispersion relation of the lattice ZK equation leads to a singular manifold, on which not only 3-wave interactions but also all m-wave interactions are allowed to happen. This phenomenon has been first observed by Lukkarinen  as a counterexample for which one of the main tools to derive kinetic equations from wave equations (the suppression of crossings) fails to hold true.