= Prasad-Sommerfield radial monopole solution
{c}
{title2=$K(\rho)=\rho/\sinh\rho,\quad H(\rho)=\coth\rho-1/\rho$}
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 $1/\rho^2-\operatorname{csch}^2\rho=(1-K^2)/\rho^2$. 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.
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