 Gain-loss Madelung equations

For the pumped Gross–Pitaevskii equation with gain term $i(\gamma -\Gamma n)\psi$, put $\psi =\sqrt{n}\exp (im\varphi /\hslash )$. The physical superfluid velocity is $\nabla \varphi$. The imaginary part gives the displayed number balance, while the real part gives $m{\varphi }_t+m|\nabla \varphi {|}^2/2+gn-{\hslash }^2{\nabla }^2\sqrt{n}/(2m\sqrt{n})=0$. 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.