= Hard-wall condensate healing profile
{title2=$n(y)=n_0\tanh^2[y/(\sqrt2\ell_h)]$}
A repulsive condensate next to an impenetrable planar wall has <Dirichlet boundary condition> $\psi(0)=0$. Writing $\psi=\sqrt{n_0}f$ reduces its stationary <Gross–Pitaevskii equation> to $\ell_h^2f''+(1-f^2)f=0$. The bulk conditions $f\to1,f'\to0$ give $\ell_h^2(f')^2=(1-f^2)^2/2$ and hence the displayed <number density>. Omitting $n_0$ describes normalized <number density>, not dimensional <number density>.
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