Phi-four kink
= Phi-four kink
{title2=$\phi^4$ kink}
For the static energy
$$
E[\phi]=\frac12\int_{-\infty}^{\infty}
\left[\phi'^2+(c^2-\phi^2)^2\right]dx,
$$
the kink joining the vacua $-c$ and $c$ is
$$
\phi(x)=c\tanh(c(x-x_0)).
$$
It saturates the <Bogomolny bound> $E\geq4c^3/3$ by satisfying the first-order equation $\phi'=c^2-\phi^2$.