Uniform-current pumped polariton condensate ($n_{\ast }=\gamma /\Gamma ,\mu =m{v}^2/2+g{n}_{\ast }$)

For positive gain and nonlinear-loss coefficients, a constant-density exciton-polariton condensate with physical velocity $v\hat{x}$ has $\psi =\sqrt{\gamma /\Gamma }\exp [i(mvx-\mu t)/\hslash ]$. 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 $\lambda (\lambda +2\gamma /\hslash )+{ϵ}_k({ϵ}_k+2g{n}_{\ast })/{\hslash }^2=0$, where ${ϵ}_k={\hslash }^2{k}^2/(2m)$. Repulsive coupling has no growing modes in this simple model; attractive coupling has a long-wavelength instability even though the plane wave exists.