Statistical equilibrium states for the nonlinear Schrödinger equation

We review a recent mean-field statistical model of self-organization in a generic class of focusing, nonintegrable nonlinear Schrödinger (NLS) equations. Such equations provide natural prototypes for nonlinear dispersive wave turbulence. The main conclusion of the theory is that the statistically preferred state for such a system is a macroscopic solitary wave coupled with fine-scale turbulent fluctuations. The coherent solitary wave is a minimizer of the Hamiltonian for a fixed particle number (or L2 norm squared), and the kinetic energy contained in the fluctuations is equipartitioned over wave numbers. Numerical simulations of the NLS equation are performed to test the predictions of the statistical model. It is demonstrated that the model accurately describes both the coherent structure and the spectral properties of the solution of the NLS system in the long-time limit.