Perturbative hydrodynamic Gross–Pitaevskii treatment for Bose–Einstein condensate in infinite length ring with disorder

For the derivation of the dilute Bose–Einstein condensate density and its phase, we have developed the perturbative approach for the solution of the stationary state couple Gross-Pitaevskii hydrodynamic equations. The external disorder potential is considered as a small parameter in this approach. We have derived expressions for the total density, condensate density, condensate density depletion and superfluid velocity of the Bose–Einstein condensate in an infinite length ring with disorder potential having a general form. For the delta correlated disorder, the explicit analytical forms of these quantities (except the superfluid velocity) have been obtained.