A statistical limit in the solution of the nonlinear Schrödinger equation under deterministic initial conditions

We investigate the semiclassical limit for the nonlinear Schrödinger equation in the case of a defocusing medium under oscillating nonperiodic initial conditions specified on the entire x axis. We formulate a system of integral conservation laws in terms of an infinite number of spatially averaged densities explicitly calculated from the initial conditions. We study the direct scattering problem and show that the scattering phase is a uniformly distributed random variable. The evolution of this system leads to the development of nonlinear oscillations, which become statistical in nature on long time scales. A modified inverse scattering method based on constructing a maximizer of the N-soliton solution in the continuum limit for N → is used to obtain an asymptotic solution. Using the maximizer, we found an infinite set of conserved averaged densities in the statistical state. This allowed us to couple the initial state with the limiting statistical steady (for t → ∞) state and, thus, to unambiguously determine the level spectrum in the statistical limit.