 Square completion for a one-dimensional kink

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.