= Gain-loss Madelung equations
{title2=$\partial_tn+\nabla\cdot(n\nabla\phi)=2n(\gamma-\Gamma n)/\hbar$}
For the pumped <Gross–Pitaevskii equation> with gain term $i(\gamma-\Gamma n)\psi$, put $\psi=\sqrt n\exp(im\phi/\hbar)$. The physical <superfluid velocity> is $\nabla\phi$. The imaginary part gives the displayed number balance, while the real part gives $m\phi_t+m|\nabla\phi|^2/2+g n-\hbar^2\nabla^2\sqrt n/(2m\sqrt n)=0$. These <Madelung equations> hold away from density zeros. The source term permits stationary density with nonzero current divergence; local gain need not balance local loss pointwise.
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