  We exhaustively construct instanton solutions and elucidate their properties
in one-dimensional anti-ferromagnetic chiral magnets based on the $O(3)$
nonlinear sigma model description of spin chains with the Dzyaloshinskii-Moriya
(DM) interaction. By introducing an easy-axis potential and a staggered
magnetic field, we obtain a phase diagram consisting of ground-state phases
with two points (or one point) in the easy-axis dominant cases, a helical
modulation at a fixed latitude of the sphere, and a tricritical point allowing
helical modulations at an arbitrary latitude. We find that instantons (or
skyrmions in two-dimensional Euclidean space) appear as composite solitons in
different fashions in these phases: temporal domain walls or wall-antiwall
pairs (bions) in the easy-axis dominant cases, dislocations (or phase slips)
with fractional instanton numbers in the helical state, and isolated instantons
and calorons living on the top of the helical modulation at the tricritical
point. We also show that the models with DM interaction and an easy-plane
potential can be mapped into those without them, providing a useful tool to
investigate the model with the DM interaction.
