Exact solution of the Thomas-Fermi equation for a trapped Bose-Einstein condensate with dipole-dipole interactions

We derive an exact solution to the Thomas-Fermi equation for a Bose-Einstein condensate (BEC) which has dipole-dipole interactions as well as the usual s-wave contact interaction, in a harmonic trap. Remarkably, despite the nonlocal anisotropic nature of the dipolar interaction the solution is an inverted parabola, as in the pure s-wave case, but with a different aspect ratio. We explain in detail the mathematical tools necessary to describe dipolar BECs with or without cylindrical symmetry. Various properties such as electrostriction and stability are discussed.