In many nonlinear systems, out-of-equilibrium states emerge when dissipation and injection of an invariant, typically energy, occur at different scales. These states are often characterized by a constant flux of the invariant across scales in a cascade process. A direct cascade refers to the transfer of an invariant from large to small scales, while an inverse cascade operates in the opposite direction. Such cascades are central to hydrodynamic and wave turbulence. In hydrodynamics, they are driven by vortex interactions, whereas in wave turbulence (WT), they result from interactions of random waves. Examples of WT in nature include turbulence of inertial and internal waves in rotating stratified fluids, gravitational waves, Kelvin waves in superfluid vortices, and Bose-Einstein condensates (BECs). Unlike hydrodynamic turbulence, where many predictions remain phenomenological, weak WT theory provides analytical predictions for the wave excitation spectrum, which can be derived as exact solutions of the wave kinetic equation (WKE).

Recent experiments with BECs have successfully demonstrated controlled WT processes in direct energy cascades. In addition to being a fascinating quantum state, BECs serve as an excellent platform for turbulence experiments, both vortex and wave-driven. This is largely due to the analogy between BEC dynamics and classical fluid flow, along with the flexibility offered by modern optical techniques, which allow for greater control than classical fluid experiments. The Gross-Pitaevskii equation (GPE), which governs BEC dynamics, is a universal nonlinear model relevant to a wide range of physical systems, including optics, plasmas, and water waves.

This talk will explore the forefront research on wave turbulence in BECs, emphasizing the universal scaling states that define fundamental aspects of the field. We will explore stationary Kolmogorov-Zakharov (KZ) spectra for free matter waves (in the 4-wave regime) and Bogoliubov waves (in the 3-wave regime) with a strong background condensate, essential for understanding energy and particle cascades in quantum systems. We also focus on self-similar evolving states in the 4-wave regime, which reflect the dynamic, non-stationary nature of wave turbulence in BECs. Through a combination of theoretical predictions and high-resolution numerical simulations, this talk will illuminate the complex behaviors of wave turbulence, providing new insights for both theoretical and experimental research in quantum fluids.

wave turbulence, superfluids, Bose gases, Bose-Einstein condensates, weak turbulence, nonlinear waves