Wave turbulence and Bose-Einstein condensates

The asymptotic behavior of a class of nonlinear Schrödinger equations is studied. Particular cases of 1D weakly focusing and Bose-Einstein condensates are considered. A statistical approach is presented following Jordan and Josserand (Phys. Rev. E 61 (2000) 1527-1539) to describe the stationary probability density of a discretized finite system. Using a maximum entropy argument, the theory predicts that the statistical equilibrium is described by energy equivalued fluctuation modes around the coherent structure minimizing the Hamiltonian of the system. Good quantitative agreement is found with numerical simulations. In particular, the particle number spectral density follows an effective 1/k² law for the asymptotic large time averaged solutions. Transient dynamics from a given initial condition to the statistically steady regime show rapid oscillations of the condensate.