Universal self-similar evolution of two-dimensional Bose-Einstein condensates in the acoustic regime

When driven out of equilibrium, a Bose-Einstein condensate develops nonlinearly interacting density waves that trigger a turbulent cascade, transferring energy toward small scales. In this article, we investigate the nonstationary evolution of solutions to the two-dimensional Gross-Pitaevskii equation. Through numerical simulations of both the GPE and the corresponding Wave Kinetic Equation, we identify self-similar solutions relevant to atomic and polariton Bose-Einstein Condensates. These solutions exhibit characteristics of both first and second kind self-similarity. In particular, we show that the dynamics of the propagating front is universal, governed by a dimensionless universal constant $\beta$, which we determine numerically.