Effective Dynamics of the Nonlinear Schrödinger Equation on Large Domains

We consider the nonlinear Schrödinger (NLS) equation posed on the box [0,L]^d with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when L is large and the characteristic size ɛ of the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in the theory of statistical physics of dispersive waves, which goes by the name of “wave turbulence.” Our main result is deriving a new equation, the continuous resonant (CR) equation, which describes the effective dynamics for large L and small ɛ over very large timescales. Such timescales are well beyond the (a) nonlinear timescale of the equation, and (b) the euclidean timescale at which the effective dynamics are given by (NLS) on ℝ^d. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of the Hardy-Littlewood circle method, which are modified and extended to be applicable in a PDE setting.