Singular charge endpoint of a complex sine-Gordon soliton ($Q(0^{\pm })=\pm 2\pi /{\lambda }^2$)

At zero rotation the field reaches ${\lambda }^2|\psi {|}^2=1$ at its center. Its energy remains finite, but its charge expression is singular and has the two displayed one-sided limits. The regular local branch excludes this endpoint. A global completion must say whether and how these two limiting charge labels represent a physical state.

An additional integer-level completion identifies the charge labels modulo $k$ and has $k-1$ nonzero sectors. For even $k$, the two maximal labels describe one self-conjugate sector. This interpretation is specified in Dorey and Hollowood, section 2; it is additional to the local classical density.