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Write $F_{12}={\partial }_1{A}_2-{\partial }_2{A}_1$. Up to a total derivative, the covariant-gradient identity and completion of the magnetic and potential terms giveFor positive flux, equality in the Bogomolny bound therefore gives the Bogomolny vortex equationsThe opposite simultaneous signs describe negative winding.

For the orientation $d{x}^1\wedge d{x}^2>0$, the Hodge star operator isThus

A radial vortex of winding $N>0$ haswith $f(0)=a(0)=0$, $f(1)=1$, $a(1)=N$, and equations

For $e^{2\rho }=8/(1-|z{|}^2{)}^2$, put $u=\log |\Phi |$. Away from zeros, the first vortex equation givesSincethe field $\psi =u-\log (1-z\bar{z})+\log 2$ satisfies the Liouville equationFor holomorphic $g:D\to D$, direct use of the Cauchy-Riemann equations verifiesChoosing $g(z)=z^{N+1}$ gives the radial Witten hyperbolic vortexFor $N=1$ these reduce toSince $Bd{\mu }_g=dA=a_1^{\prime }(r)dr\wedge d\theta$,This directly verifies the Abelian Higgs vortex flux relation for unit winding.

On the sector $0\le \varphi \le 1$, choose the superpotentialThe static energy has the Bogomolny bound completionEquality holds for ${\varphi }^{\prime }=W^{\prime }$, and therefore

For the static energythe kink joining the vacua $-c$ and $c$ isIt saturates the Bogomolny bound $E\ge 4c^3/3$ by satisfying the first-order equation ${\varphi }^{\prime }=c^2-{\varphi }^2$.

A sigma-model lump is a finite-energy classical field-theory soliton of the two-dimensional O3 nonlinear sigma model. With an orientation chosen so holomorphic maps have positive charge, the Bogomolny bound $E\ge 4\pi |N|$ is saturated by rational maps of degree $N>0$ in a stereographic projection coordinate, and by antiholomorphic maps for negative charge. The field remains smooth at coordinate poles, which are points at the other pole of the target sphere. Scale invariance permits lumps of arbitrary size; topology alone does not prevent concentration to a singular limiting configuration.

For the static energy $E=\int [{\varphi }^{\prime 2}+W^{\prime }(\varphi {)}^2]dx/2$, completing the square givesThe Bogomolny bound is the absolute boundary difference. Equality requires the Bogomolny equation ${\varphi }^{\prime }=\pm W^{\prime }(\varphi )$ with the sign selected by that difference. Differentiating it gives the Euler-Lagrange field equation ${\varphi }^{\prime \prime }=W^{\prime }{W}^{\prime \prime }$. A stationary solution of that second-order equation need not saturate the bound in a general field theory.

For $T=-[\cos \varphi {]}_{-\infty }^{+\infty }/2$, complete the static energy into squares $({\varphi }_x\mp m\sin \varphi {)}^2$ and ${\sin }^2\varphi {\alpha }_x^2$. The boundary term is $\pm rmT$. A $T=1$ saturating kink is $\varphi =2\arctan e^{m(x-x_0)}$, with constant $\alpha$ and mass $rm$. This is a model-specific Bogomolny bound.