A mean-field statistical theory for the nonlinear Schrodinger equation

A statistical model of self-organization in a generic class of one-dimensio nal nonlinear Schrodinger (NLS) equations on a bounded interval is develope d. The main prediction of this model is that the statistically preferred st ate for such equations consists of a deterministic coherent structure coupl ed with fine-scale, random fluctuations, or radiation. The model is derived from equilibrium statistical mechanics by using a mean-field approximation of the conserved Hamiltonian and particle number (L-2 norm squared) for fi nite-dimensional spectral truncations of the NLS dynamics. The continuum li mits of these approximated statistical equilibrium ensembles on finite-dime nsional phase spaces are analyzed, holding the energy and particle number a t fixed, finite values. The analysis shows that the coherent structure mini mizes total energy for a given value of particle number and hence is a solu tion to the NLS ground state equation, and that the remaining energy reside s in Gaussian fluctuations equipartitioned over wave numbers. Some results of direct numerical integration of the NLS equation are included to validat e empirically these properties of the most probable states for the statisti cal model. Moreover, a theoretical justification of the mean-field approxim ation is given, in which the approximate ensembles are shown to concentrate on the associated microcanonical ensemble in the continuum limit. (C) 2000 Elsevier Science B.V. All rights reserved.