- i\frac{\gamma}{2}\Big)\psi_0   + \Omega\\\nonumber
&& + \frac{2U\psi_0}{L}\sum_k n_k + \frac{U\psi_0^\ast}{L}\sum_k c_k + \frac{U}{\sqrt{L^3}} \sum_{k,q} M^\ast_{k,q}.
\label{eq:HE_MF}\end{aligned}$$ The second-order correlation functions (\[eq:2nd\_bog\]) are now coupled to the third-order correlation functions (\[eq:3rd\_corrs\]) $$\begin{aligned}
\nonumber
\label{eq:HE_n}
i\partial_t n_k &=& -i\gamma n_k +  2i{\text{Im}}\bigg[ U\Big(\psi_0^2 + \frac{1}{L}\sum_q c_q \Big) c^\ast_k \\
&& + \frac{2U\psi_0}{\sqrt{L}} \sum_q  M_{q,k} + \frac{U\psi_0^\ast}{\sqrt{L}} \sum_q M^\ast_{k,q} \bigg],\\
\nonumber
\label{eq:HE_c}
i\partial_t c_k &=& \bigg(2\epsilon_k+2U\Big( |\psi_0|^2 +\frac{1}{L} \sum_q n_q\Big) -i\gamma\bigg) c_k \\\nonumber
&&+ U\Big( \psi_0^2 + \frac{1}{L} \sum_q c_q \Big)(2 n_k + 1)\\\nonumber
&&+\frac{2U\psi_0}{\sqrt{L}}\sum_q\left(M_{q,-k}^\ast + M_{q,k}^\ast\right) \\
&&+ \frac{U\psi_0^\ast}{\sqrt{L}}\sum_q \left( R_{-k,q } + R_{k,q} \right).\end{aligned}$$ Note that the factorized contribution of the fourth-order correlator enters here in the equations of motion as a small correction to the couplings $\psi_0^2$ and $|\psi_0|^2$. At equilibrium these corrections are well-studied in the Hartree-Fock-Bogoliubov method [@andersen].

Finally we also find the equations of motion for the third-order correlation functions, in which the fifth-order back-reaction is neglected $$\begin{aligned}
\nonumber
i\partial_t M_{k,q} &=& \left( \epsilon_k - \epsilon_q  - \epsilon_{k-q} - U|\psi_0|^2-\frac{3i}{2}\gamma \right) M_{k,q} \\\nonumber
&&- U\big(\psi_0^\ast\big)^2\left(M^\ast_{q,k} + M^\ast_{k-q,k} \right) \\
&&+ U\psi_0^2\; R^\ast_{-k,q} +  F^{(M)}_{k,q} , \label{eq:HE_M}\\
\nonumber
i\partial_t R_{k,q} &=& \left( \epsilon_k +\epsilon_q + \epsilon_{k+q} + 3U|\psi_0|^2 -\frac{3i}{2}\gamma\right) R_{k,q} \\\nonumber
&&+ 
U\psi_0^2\left(M^\ast_{-k,q}+M^\ast_{-q,k} + M^\ast_{k+q,k} \right)\\
&& + 