  Domain wall skyrmions are skyrmions trapped inside a domain wall. We
investigate domain wall skyrmions in chiral magnets using a fully analytic
approach. Treating the Dzyaloshinskii-Moriya (DM) interaction perturbatively,
we construct the low-energy effective theory of a magnetic domain wall in an
$O(3)$ sigma model with the DM interaction and an easy-axis potential term,
yielding a sine-Gordon model. We then construct domain wall skyrmions as
sine-Gordon solitons along the domain wall. We also construct domain wall
skyrmions on top of a pair of a domain wall and an anti-domain wall. One of
characteristic feature of domain wall skyrmions is that both skyrmions and
anti-skyrmions are equally stable inside the domain wall, unlike the bulk in
which only one of them is stable.
