= Solution
A <Bogomolny bound> expresses a static field energy as nonnegative squares plus a term fixed by the <topological charge> or boundary data. Setting the squares to zero gives the first-order <Bogomolny equations>. Their solutions minimize the energy in that sector and satisfy the second-order <Euler-Lagrange field equations>, although a general stationary solution need not attain the bound.
For one real scalar in one spatial dimension, take the <Lagrangian density>
$$
\mathcal L=\frac12\dot\phi^2-\frac12\phi_x^2-\frac12[W'(\phi)]^2.
$$
A <finite-energy field configuration> in this static sector must approach <scalar-field vacua> $\phi_\pm$ with $W'(\phi_\pm)=0$ at the two ends. <Completing the square> gives the <square completion for a one-dimensional kink>:
$$
E=\frac12\int_{\mathbb R}(\phi_x\mp W')^2dx\pm\int_{\mathbb R}\phi_xW'(\phi)dx
=\frac12\int_{\mathbb R}(\phi_x\mp W')^2dx\pm[W(\phi_+)-W(\phi_-)].
$$
Choose the sign for which the boundary term is nonnegative. Thus
$$
\boxed{E\geq|W(\phi_+)-W(\phi_-)|,\qquad \phi_x=\pm W'(\phi)\text{ at equality}.}
$$
Taking the <derivative> of the equality equation gives $\phi_{xx}=W'W''=d([W']^2/2)/d\phi$, the static <Euler-Lagrange field equation>. The boundary term is invariant under deformations keeping the asymptotic <scalar-field vacua> fixed; it is not a contribution from the local shape of the kink.
For a <phi-four kink>, let $c>0$, $W(\phi)=c^2\phi-\phi^3/3$, and select the sector $\phi_-=-c$, $\phi_+=c$. Then $W'=c^2-\phi^2$, and the increasing <Bogomolny equation> is $\phi_x=c^2-\phi^2$. Separating variables yields
$$
\frac1{2c}\log\frac{c+\phi}{c-\phi}=x-X,
\qquad
\boxed{\phi_K(x)=c\tanh(c(x-X)),\qquad M=\frac{4c^3}{3}.}
$$
The integration constant $X$ is the translational <collective coordinate>. Directly, $\phi_K'=c^2\operatorname{sech}^2(c(x-X))$ and its <energy density> is $c^4\operatorname{sech}^4(c(x-X))$, whose integral is $4c^3/3$. The decreasing antikink has the reversed boundary values, profile $-\phi_K$, and the same energy. A <Lorentz boost> produces the exact uniformly moving kink $c\tanh[c\gamma(x-X_0-vt)]$, with <Lorentz factor> $\gamma=(1-v^2)^{-1/2}$ and energy $\gamma M$.
In two spatial dimensions, choose the <Abelian Higgs model> at <critical coupling>, with $D_i=\partial_i-iA_i$, $B=\partial_1A_2-\partial_2A_1$, and energy
$$
E=\int_{\mathbb R^2}\left[\frac12B^2+\frac12(|D_1\phi|^2+|D_2\phi|^2)+\frac18(1-|\phi|^2)^2\right]d^2x.
$$
The <gauge covariant derivative> is used throughout this normalization. <Integration by parts>, using $[D_1,D_2]=-iB$, gives $\int|D_i\phi|^2=\int|(D_1+iD_2)\phi|^2+\int B|\phi|^2$, with a vanishing boundary divergence for the decaying vortex fields. Combining this identity with the magnetic and potential terms gives the <Bogomolny square completion for an Abelian Higgs vortex>:
$$
E=\int\left[\frac12|(D_1+iD_2)\phi|^2+\frac12\left(B-\frac{1-|\phi|^2}{2}\right)^2\right]d^2x+\frac12\int B\,d^2x.
$$
<Finite-energy field configurations> have $|\phi|\to1$ at infinity, and $D_i\phi\to0$ makes the <magnetic flux> equal to the phase winding $2\pi N$. For $N>0$,
$$
\boxed{E\geq\pi N,\qquad(D_1+iD_2)\phi=0,\qquad B=\frac{1-|\phi|^2}{2}.}
$$
These are the <Bogomolny vortex equations> for an <Abelian Higgs vortex>. Opposite signs give antivortices and the bound $E\geq\pi|N|$. The coefficient $\pi$ follows from the explicit energy normalization above; other conventions can give $2\pi$. Static solutions of fixed positive <vortex number> have equal energy independent of their positions, giving the <Abelian Higgs vortex moduli space> used for slow dynamics.
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