= Reservoir reduction of a polariton condensate
For reservoir kinetics $\mathcal R_t=-(\gamma_R+R_Rn)\mathcal R+P$, <adiabatic elimination> gives $\mathcal R=P/(\gamma_R+R_Rn)$. Expanding for $R_Rn/\gamma_R\ll1$ and inserting into the condensate equation gives a <complex Ginzburg–Landau equation> with
$$
\alpha=\frac12\left(\frac{R_RP}{\gamma_R}-\gamma_C\right),\quad
\beta=\frac{R_R^2P}{2\gamma_R^2},\quad
g=U_0-\frac{g_RR_RP}{\gamma_R^2},\quad s=-\frac{g_RP}{\gamma_R}.
$$
Fast reservoir response and weak depletion are separate assumptions. Without the <number density> expansion the gain is saturable, not cubic. The coefficients depend on position when the pump does.
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