the generalized Gross-Pitaevskii equation. At the level of approximation considered in this section, both approaches are equivalent, but setting $\phi_0 = 0$ asserts that we are dealing with connected second- and third-order correlation functions, which is a better controlled truncation [@kohler_microscopic]. We refer the interested readers to Appendix \[app:HOC\] for more details on the method and to Appendix \[app:comp\] for a comparison of different truncation schemes.

Correlation functions of order four, which enter into the equations of motion of the second and third-order correlators, are factorized into different possible products of second-order correlation functions. With this procedure we explicitly neglect the connected part of the fourth-order correlation function, but we keep its main contribution coming from separable correlations. The fifth-order correlator, entering in the equation of motion for the third-order correlation functions, can be instead safely neglected. Already in factorized form it would reduce to various products of second and third order, which constitute negligible corrections to dominant terms in the equations of motion. See Appendix \[app:HOC\] for more details on the implications of these approximations.

Within this framework, the Gross-Pitaevskii equation for the homogeneous condensate background $\psi_0$, extended with the back-reaction terms reads $$\begin{aligned}
i \partial_t \psi_0 &=& \Big(-\Delta 