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Classical and quantum field theory of dipolar, axisymmetric quadrupolar andoctupolar Bose gases is considered within a general approach. Dipole,axisymmetric quadrupole and octupole interaction potentials in the momentumrepresentation are calculated. These results clearly demonstrate attraction andrepulsion areas in corresponding gases. Then the Gross-Pitaevskii (GP)equation, which plays a key role in the present paper, is derived from thecorresponding functional. The zoology of the form factors appearing in GPequation is studied in details. The classes of proper for the description ofspatially non-uniform condensates form factors are chosen. In the Thomas-Fermiapproximation a general solution of the GP equation with a quasilocal formfactor is obtained. This solution has an interesting form in terms of a doublerapidly converging series that universally includes all the interactionsconsidered. An important analysis of the condensate stability, in other wordsthe study of condensate excitations, is also performed in this paper. In theGaussian approximation (from the GP functional), a functional describing theperturbations of the condensate is derived in details. For a probe wavefunction in the form of a plane wave, a spectrum of Bogoliubov excitations wasobtained, from which an equation describing the threshold momentum for theemergence of instability was derived. An important result of this paper is thedependence of the threshold on the momentum of a stationary condensate. Forcompleteness of the presentation, the approximating expression in the form of arapidly converging series is obtained for the corresponding dependence.Finally, in the conclusion a quantum hydrodynamic theory for dipolar,axisymmetric quadrupolar and octupolar gases is briefly presented, giving aclue to the experimental determination of the form factors.

Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross-Pitaevskii Formalism with Form Factor