# Effective dynamics of the nonlinear Schroedinger equation on large domains

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We consider the nonlinear Schroedinger equation posed on a large box of characteristic size $L$, and ask about its effective dynamics for very long time scales. After pointing out some “more or less” trivial time scales along which the effective dynamics can be easily described, we start inspecting some much longer time scales where we notice some non-trivial dynamical behaviors. Particularly, the end goal of such an analysis is to reach the so-called “kinetic time scale”, at which it is conjectured that the effective dynamics is governed by a kinetic equation called the “wave kinetic equation”. This is the subject of wave turbulence theory. We will discuss some recent advances towards this end goal. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah.