= Cubic-quintic quantum vortex tail
{title2=$R(r)=1-\mathcal N^2/[2(1+\beta)r^2]+O(r^{-3})$}
For a <cubic-quintic Gross–Pitaevskii equation> with normalized bulk amplitude one, a straight <quantum vortex> of integer <winding number> $\mathcal N$ has radial equation $R''+R'/r-\mathcal N^2R/r^2+(1-\alpha R^2-\beta R^4)R=0$. With $\alpha+\beta=1$, the derivative of the nonlinear term at $R=1$ is $-2(1+\beta)$. Balancing the order-$r^{-2}$ centrifugal term with the bulk amplitude restoring term proves the displayed tail for a stable nondegenerate bulk state, $1+\beta>0$. A vortex's amplitude tends to one; its full complex field tends to $e^{i\mathcal N\theta}$, not a direction-independent constant. At $\beta=-1$ the stated inverse-square expansion cannot cancel the centrifugal term.
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