Wave turbulence and vortices in Bose-Einstein condensation

We report a numerical study of turbulence and Bose-Einstein condensation within the two-dimensional Gross-Pitaevsky model with repulsive interaction. In the presence of weak forcing localized around some wave number in the Fourier space, we observe three qualitatively different evolution stages. At the initial stage a thermodynamic energy equipartition spectrum forms at both smaller and larger scales with respect to the forcing scale. This agrees with predictions of the four-wave kinetic equation of the Wave Turbulence (WT) theory. At the second stage, WT breaks down at large scales and the interactions become strongly nonlinear. Here, we observe formation of a gas of quantum vortices whose number decreases due to an annihilation process helped by the acoustic component. This process leads to formation of a coherent-phase Bose-Einstein condensate. After such a coherent-phase condensate forms, evolution enters a third stage characterised by three-wave interactions of acoustic waves that can be described again using the WT theory.