Inverse-square tail of a Gross–Pitaevskii vortex
= Inverse-square tail of a Gross–Pitaevskii vortex
{title2=$f(r)=1-\mathcal N^2/(2r^2)+O(r^{-4})$}
In healing-length units a straight <quantum vortex> obeys $f''+f'/r-\mathcal N^2f/r^2+(1-f^2)f=0$ and $f\to1$. Inserting $f=1-a/r^2+\cdots$ makes its order-$r^{-2}$ balance $-\mathcal N^2+2a=0$, proving the tail. The physical <healing length> convention must be stated; the deficit of density $1-f^2$ has twice the leading coefficient of the amplitude deficit $1-f$.