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Sustained turbulence in the three-dimensional Gross-Pitaevskii model

We study the 3D forced-dissipated Gross-Pitaevskii equation. We force at relatively low wave numbers, expecting to observe a direct energy cascade and a consequent power-law spectrum of the form $k^{-α}$. Our numerical results show that the exponent $α$ strongly depends on how the inverse particle cascade is attenuated at $k$'s lower than the forcing wave number. If the inverse cascade is arrested by a friction at low $k$'s, we observe an exponent which is in good agreement with the weak wave turbulence prediction $k^{-1}$. For a hypo-viscosity, a $k^{-2}$ spectrum is observed which we explain using a critical balance argument. In simulations without any low-$k$ dissipation, a condensate at $k=0$ is growing and the system goes through a strongly-turbulent transition from a four-wave to a three-wave weak turbulence acoustic regime with $k^{-3/2}$ Zakharov-Sagdeev spectrum. In this regime, we also observe a spectrum for the incompressible kinetic energy which formally resembles the Kolmogorov $k^{-5/3}$, but whose correct explanation should be in terms of the Kelvin wave turbulence. The probability density functions for the velocities and the densities are also discussed.