Bogoliubov-wave turbulence in Bose-Einstein condensates

We theoretically and numerically study Bogoliubov-wave turbulence in three-dimensional atomic Bose-Einstein condensates with the Gross-Pitaevskii equation, investigating three spectra for the macroscopic wave function, the density distribution, and the Bogoliubov-wave distribution. In this turbulence, Bogoliubov waves play an important role in the behavior of these spectra, so that we call it Bogoliubov-wave turbulence. In a previous study [D. Proment \textit{et al.}, Phys. Rev. A \textbf{80}, 051603(R) (2009)], a $-3/2$ power law in the spectrum for the macroscopic wave function was suggested by using weak wave turbulence theory, but we find that another $-7/2$ power law appears in both theoretical and numerical calculations. Furthermore, we focus on the spectrum for the density distribution, which can be observed in experiments, discussing the possibility of experimental observation. Through these analytical and numerical calculations, we also demonstrate that the previously neglected condensate dynamics induced by the Bogoliubov waves is remarkably important.