Statistical Mechanics of the Nonlinear Schr6dinger Equation. II. Mean Field Approximation

We investigate a mean field approximation to the statistical mechanics of complex fields with dynamics governed by the nonlinear Schr6dinger equation. Such fields, whose Hamiltonian is unbounded below, may model plasmas, lasers, and other physical systems. Restricting ourselves to one-dimensional systems with periodic boundary conditions, we find in the mean field approximation a phase transition from a uniform regime to a regime in which the system is dominated by solitons. We compute explicitly, as a function of temperature and density (L 2 norm), the transition point at which the uniform configuration becomes unstable to local perturbations; static and dynamic mean field approximations yield the same result.