Prasad-Sommerfield radial monopole solution ($K(\rho )=\rho /\sinh \rho ,H(\rho )=\coth \rho -1/\rho$)

For a Bogomolny-Prasad-Sommerfield monopole, the dimensionless radial Bogomolny equations are $dK/d\rho =-KH$ and $dH/d\rho =(1-K^2)/{\rho }^2$. The displayed profiles solve both equations: differentiating $\rho /\sinh \rho$ gives $K(1/\rho -\coth \rho )=-KH$, while differentiating $H$ gives $1/{\rho }^2-{\mathrm{csch}}^2\rho =(1-K^2)/{\rho }^2$. At the origin $K=1-{\rho }^2/6+O({\rho }^4)$ and $H=\rho /3+O({\rho }^3)$, giving a regular core. At infinity $K$ decays exponentially and $H=1-1/\rho +O(e^{-2\rho })$, giving the vacuum Higgs magnitude and a long-range Abelian magnetic field.