  We present a statistical equilibrium model of self-organization in a class of
focusing, nonintegrable nonlinear Schrodinger (NLS) equations. The theory
predicts that the asymptotic-time behavior of the NLS system is characterized
by the formation and persistence ofa large-scale coherent solitary wave, which
minimizes the Hamiltonian given the conserved particle number,coupled with
small-scale random fluctuations, or radiation. The fluctuations account for the
difference between the conserved value of the Hamiltonian and the Hamiltonian
of the coherent state. The predictions of the statistical theory are tested
against the results of direct numerical simulations of NLS, and excellent
qualitative and quantitative agreement is demonstrated. In addition, a careful
inspection of the numerical simulations reveals interesting features of the
transitory dynamics leading up to the to the long-time statistical equilibrium
state starting from a given initial condition. As time increases, the system
investigates smaller and smaller scales, and it appears that at a given
intermediate time after the coalescense of the soliton structures has ended,
the system is nearly in statistical equilibrium over the modes that it has
investigated up to that time.
