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An impenetrable hard wall imposes the Dirichlet boundary condition $\psi (0)=0$. Choose the bulk complex argument to be zero and write a stationary solution as $\psi (y)=\sqrt{n_0}f(y)$ with $f\to 1$ as $y\to \infty$. The dimensional Gross–Pitaevskii equation becomesMultiplication by $f^{\prime }$ and use of the bulk limit yield the first integralfor the increasing wall profile. Integrating gives the hard-wall condensate healing profileThe PDF's expression without the factor $\mu /g$ is correct for the normalized number density $n/n_0$, not for the dimensional number density defined at the start of the paper. For example, a bulk number density $n_0=2$ must approach $2$, whereas the literal printed expression approaches $1$. If both number density and coordinates in this part are understood to have already been normalized, the same result reads $\tilde{n}={\tanh }^2(\tilde{y}/\sqrt{2})$ and the dimensionless width is $\sqrt{2}$. These are two descriptions of the same profile, not different physical healing lengths.