= Solution
Let $n=|\Psi|^2$. Two independent approximations are required for the <reservoir reduction of a polariton condensate>. First the <exciton> reservoir must follow the changing condensate <number density> rapidly: after its initial transient, its relaxation time $[\gamma_R+R_Rn]^{-1}$ must be small compared with the local density-evolution time. A sufficient modewise condition is that the relevant <number density> frequencies and growth rates are much smaller than $\gamma_R+R_Rn$; pump variations must be comparably slow. There is no reservoir diffusion in the given model. <Adiabatic elimination> then gives
$$
\mathcal R\simeq\frac{P(\mathbf r)}{\gamma_R+R_Rn}.
$$
Second, to obtain the particular cubic equation by expansion about the empty condensate, require $\varepsilon=R_Rn/\gamma_R\ll1$. Then
$$
\mathcal R=\frac P{\gamma_R}\left(1-\frac{R_R}{\gamma_R}n\right)+O\!\left(\frac P{\gamma_R}\varepsilon^2\right).
$$
Keeping the full denominator instead gives a saturable-gain equation, not exactly the requested cubic <complex Ginzburg–Landau equation>. A near-threshold condensate is one natural regime in which the second condition holds. Rapid reservoir relaxation alone does not justify this <number density> expansion.
Multiplying the condensate equation by $-i$ and inserting the expanded reservoir separates real gain from the conservative frequency shift:
$$
\boxed{\begin{aligned}
\alpha&=\frac12\left(\frac{R_RP}{\gamma_R}-\gamma_C\right),&
\beta&=\frac{R_R^2P}{2\gamma_R^2},\\
g&=U_0-\frac{g_RR_RP}{\gamma_R^2},&
s&=-\frac{g_RP}{\gamma_R}.
\end{aligned}}
$$
These are local functions if the pump is spatially varying. Their signs are important: the reservoir blueshift gives negative $s$ in the printed convention $+is\Psi$, while reservoir depletion subtracts from the effective interaction $g$. Positive rates and pump give $\beta>0$; the effective $g$ need not be positive merely because the original interactions are repulsive.
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