= Uniform-current pumped polariton condensate
{title2=$n_*=\gamma/\Gamma,\quad\mu=mv^2/2+g n_*$}
For positive gain and nonlinear-loss coefficients, a constant-density <exciton-polariton condensate> with physical velocity $v\widehat x$ has $\psi=\sqrt{\gamma/\Gamma}\exp[i(mvx-\mu t)/\hbar]$. Its <chemical potential> obeys the displayed relation. For prescribed real $\mu$, a real velocity requires $\mu\ge g\gamma/\Gamma$. If chemical potential is determined by the solution, any real velocity is allowed. Linear density-phase perturbations in the comoving frame satisfy $\lambda(\lambda+2\gamma/\hbar)+\epsilon_k(\epsilon_k+2g n_*)/\hbar^2=0$, where $\epsilon_k=\hbar^2k^2/(2m)$. Repulsive coupling has no growing modes in this simple model; attractive coupling has a long-wavelength instability even though the plane wave exists.
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