Preprint Open Access
Jonathan Skipp, Victor L'vov, Sergey Nazarenko
We develop the theory of weak wave turbulence in systems described by the Schrödinger-Helmholtz equation in two and three dimensions. This model contains as limits both the familiar cubic nonlinear Schrödinger equation, and the Schrödinger-Newton equations, the latter being a nonrelativistic model of Fuzzy Dark Matter which has a nonlocal gravitational self-potential. We show that in the weakly-nonlinear limit the Schrödinger-Helmholtz equation has a simultaneous inverse cascade of particles and a forward cascade of energy. The inverse cascade we interpret as a nonequilibrium condensation process, which could be a precursor to collapses and structure formation at large scales (for example the formation of galactic dark matter haloes). We show that for the Schrödinger-Newton equation in two and three dimensions, and in the two-dimensional nonlinear Schrödinger equation, the particle and energy fluxes are carried by small deviations from thermodynamic distributions, rather than the Kolmogorov-Zakharov cascades that are familiar in wave turbulence. We develop a differential approximation model to characterise such "warm cascade" states.