Energy distribution and wave turbulence closures for the nonlinear Schrodinger equation

We introduce the theory of wave turbulence as a systematic approach to studying the energy distribution across scales in nonlinear dispersive PDE. This is done by deriving an effective equation for the energy density of the system in a statistical setting, by taking weak nonlinearity and infinite volume limits. The resulting equation is called the “wave kinetic equation”, and it gives, at a formal level, a lot of insight into the out-of-equilibrium dynamics and statistics of nonlinear dispersive systems. The fundamental problem here is to give a rigorous derivation of this formally derived equation. Without any stochastic element in the system, such problems are often too difficult to resolve (even in much simpler ODE settings). We will show how this equation can be derived starting from the nonlinear Schrodinger equation on a large torus, in the presence of an appropriate random force, and in the weakly nonlinear infinite volume limit. This is joint work with Isabelle Gallagher and Pierre Germain

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