Scalar-field kink
= Scalar-field kink
= Kink
{synonym}
A scalar-field kink is a one-dimensional finite-energy configuration approaching different vacuum values at the two spatial infinities. For $V(\phi)=\lambda(\phi^2-v^2)^2/4$, the static equation is $\phi''=\lambda\phi(\phi^2-v^2)$ and has the solution
$$
\phi(x)=v\tanh\!\left[v\sqrt{\lambda/2}(x-x_0)\right].
$$
The center $x_0$ is arbitrary because of translation symmetry. Its inverse-width scale is $v\sqrt{\lambda/2}$, and substitution verifies the equation and its opposite vacuum limits.
The boundary change distinguishes a kink from a small fluctuation around one vacuum. An antikink reverses the limits; a kink and antikink together can have trivial total topological sector and need not remain separate forever. This model makes a domain wall's core and finite energy explicit without treating the entire configuration as a point particle.