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An efficient numerical method for investigating the dynamics of Bose-Fermi mixture at extremely low temperature

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Starting from the quantum theory, by means of mean-field approximation for Bosons and Wigner approximation for Fermions, we review on how to derive time-dependent Schrodinger-Vlasov (SV) system for investigating the dynamics of quantum fluids composed of a Bose-Einstein condensate and a cloudoffermionic atoms at extremely low temperature. We then discuss the mathematical properties of the SV system and propose an efficient time-splitting Fourier pseudospectral method for the SV system and apply it to study the mean-field dynamics of Bose-Fermi mixture. The numerical integration for the system is performed using the well-known splitting'' method, coupled with Fourier pseudospectral method and semi-Lagrangian method. The algorithm provides spectral accuracy in space and can be implemented efficiently with the fast Fourier transform. Interesting numerical results on the dynamics of Bose-Fermi mixture by the proposed numerical method are shown.