We perform numerical simulations of turbulent and complex flows in a Bose-Einstein condensate, using the two-dimensional defocusing stochastically-forced Gross–Pitaevskii equation. Building on the theory and results derived in Refs. [20, 21] and [25], we aim to extend the quantitative understanding of the features of turbulence. Following Ref. [21], the forcing is defined in Fourier space as a spectrally band-limited function, effectively injecting energy into the system within a given range of wavenumbers. This injection is balanced by the removal of particles/energy both at large and small-length scales, through phenomenological hypoviscosity and hyperviscosity terms. Removing the damping terms and shifting the forcing region give rise to different regimes. We report on the study of three of them, which we call weak-wave turbulence, low-frequency saturation and high-frequency saturation. The Low-Frequency Saturation and High-Frequency Saturation regimes are obtained via the removal of the hypoviscosity term, allowing for the accumulation of particles at large scales and thus condensate formation. We implement the data-driven technique of Dynamic Mode Decomposition to extract the most energetic modes from the system and characterize the complex patterns (coherent structures) that emerge in the different regimes.
