= Solution
Use the dimensionless <quantum phase> $S$, so that $\psi=R e^{iS}$ with $R\geq0$. Work locally away from <wavefunction> nodes, where $R$ and $S$ are differentiable. Differentiating the <wavefunction> for substitution in the <Time-dependent Schrodinger equation> gives:
$$
\partial_t\psi=e^{iS}(R_t+iRS_t),\qquad
\nabla^2\psi=e^{iS}\left[\nabla^2R-R|\nabla S|^2+i(2\nabla R\cdot\nabla S+R\nabla^2S)\right].
$$
Cancel $e^{iS}$ in the time-dependent equation and equate real and imaginary parts. The resulting <Madelung equations> are
$$
\boxed{R_t=-\frac{\hbar}{2m}\left(2\nabla R\cdot\nabla S+R\nabla^2S\right)},\qquad
\boxed{\hbar S_t+\frac{\hbar^2}{2m}|\nabla S|^2+V-\frac{\hbar^2}{2m}\frac{\nabla^2R}{R}=0}.
$$
The second equation is a <Hamilton-Jacobi equation> for the action $\hbar S$, with an additional <quantum potential>. To see the meaning of the first, multiply it by $2R$ and set $\rho=R^2$. It becomes the <probability continuity equation>
$$
\partial_t\rho+\nabla\cdot\left(\rho\frac{\hbar}{m}\nabla S\right)=0.
$$
Thus the <probability density> is transported by the velocity field $\hbar\nabla S/m$. The <Madelung equations> are a local rewriting of the linear wave equation; their apparent nonlinearity comes from expressing a complex <wavefunction> in modulus and phase variables.
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