  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 $k^{-\alpha}$. Our numerical results
show that the exponent $\alpha$ 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.
