 Scalar-field kink

A scalar-field kink is a one-dimensional finite-energy configuration approaching different vacuum values at the two spatial infinities. For $V(\varphi )=\lambda ({\varphi }^2-v^2{)}^2/4$, the static equation is ${\varphi }^{\prime \prime }=\lambda \varphi ({\varphi }^2-v^2)$ and has the solutionThe 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.