Quantum turbulence cascades in the Gross-Pitaevskii model

We present a numerical study of quantum turbulence within the three-dimensional Gross-Pitaevskii equation, concentrating on the direct energy cascade in the case of a forced-dissipated system. We show that the behavior of the system is very sensitive to the properties of the model at the scales greater than the forcing scale, and we identify three different regimes: (1) a nonstationary regime with condensation and transition from a four-wave to a three-wave interaction process when the largest scales are not dissipated, (2) a steady weak wave turbu- lence regime when largest scales are dissipated with a friction-type dissipation, and (3) a state with a scale- by-scale balance of the linear and the nonlinear time scales when the large-scale dissipation is a hypoviscosity.