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Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement

Ground states and dynamical properties of dipolar Bose-Einstein condensate are analyzed based on the Gross-Pitaevskii-Poisson system (GPPS) and its dimension reduction models under anisotropic confining potential. We begin with the three-dimensional (3D) Gross-Pitaevskii-Poisson system and review its quasi-2D approximate equations when the trap is strongly confined in $z$-direction and quasi-1D approximate equations when the trap is strongly confined in $x$-, $y$-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-Δ)^{1/2}$ is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both equations and finite time blowup in 2D are analyzed. Finally, we rigorously prove the convergence and linear convergence rate between the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in weak interaction regime.