Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 2 / c / Solution

For a rotationally symmetric vortex, the equation away from the origin isInsertThe prescribed zero fixes $\alpha =2N$. Since $N\ge 1$, $e^u=e^{\beta }{r}^{2N}[1+o(1)]$, whileMatching the singular and constant terms with $e^u-1=-1+o(1)$ givesThe undetermined $\beta$ is fixed by matching this local expansion to $u\to 0$ at infinity.