Integration Theory for Kinks and Sphalerons in One Dimension

The static kink, sphaleron and kink chain solutions for a single scalar field $\phi$ in one spatial dimension are reconsidered. By integration of the Euler--Lagrange equation, or through the Bogomolny argument, one finds that each of these solutions obeys a first-order field equation, an autonomous ODE that can always be formally integrated. We distinguish the BPS case, where the required integral is along a contour in the $\phi$-plane, from the semi-BPS case, where the integral is along a contour in the Riemann surface double-covering the $\phi$-plane, and is generally more complicated.