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Non-integral form of the Gross-Pitaevskii equation for polarized molecules

The Gross-Pitaevskii equation for polarized molecules is an integro-differential equation, consequently it is complicated for solving. We find a possibility to represent it as a non-integral nonlinear Schrodinger equation, but this equation should be coupled with two linear equations describing electric field. These two equations are the Maxwell equations. We recapture the dispersion of collective excitations in the three dimensional electrically polarized BEC with no evolution of the electric dipole moment directions. We trace the contribution of the electric dipole moment. We explicitly consider the contribution of the electric dipole moment in the interaction constant for the short-range interaction. We show that the spectrum of dipolar BEC reveals no instability at repulsive short-range interaction. Nonlinear excitations are also considered. We present dependence of the bright soliton characteristics on the electric dipole moment.