Nonequilibrium statistical behavior of nonlinear Schrِdinger equations

This paper explores the equilibrium and near-equilibrium statistical behavior of nonlinear Schrِdinger equations through numerical simulations. The investigation centers on a class of finite-dimensional Hamiltonian systems obtained by discretizing NLS equations defined on a bounded interval with saturated focusing nonlinearities. These nonlinear systems govern a wave dynamics that is nonintegrable and nonsingular, and for which typical solutions are composed of large-scale coherent structures interacting within a background of small-scale fluctuations. The equilibrium statistics of such a system is most naturally defined by a Gibbs distribution that is canonical in energy and microcanonical in particle number. By sampling these equilibrium distributions using a Metropolis-type algorithm, the typical excitations of the system are determined over the full range of system parameters. Nonequilibrium relaxations from initial perturbations of these equilibrium distributions are then computed by means of linear response theory. The computations reveal a surprising dependence of the qualitative properties of relaxation toward equilibrium on the temperature and the nonlinearity. On the one hand, a simple exponential decay to equilibrium is observed for sufficiently high temperature or sufficiently strong nonlinearity. For low temperature and moderate nonlinearity, on the other hand, the system rapidly adjusts to a metaequilibrium state that equilibrates extremely slowly. This two-stage thermalization may be related to quasiperiodicity in the underlying dynamics. Indeed, the metaequilibria are found to occur in a range of system parameters for which approximately quasiperiodic solutions are prevalent in equilibrium.