Josephson vortices and the Atiyah-Manton construction

We show that sine-Gordon solitons appear in the low-energy effective theory of a domain wall in a $U(1)$ gauge theory with two charged complex scalar fields with masses if we introduce the Josephson interaction term between the scalar fields. We identify these sine-Gordon solitons as vortices or CP¹ sigma model instantons in the bulk, which are absorbed into the domain wall world volume. These vortices can be called Josephson vortices since they appear in Josephson junctions of two superconductors. This setup gives a physical realization of a lower-dimensional analogue of Atiyah-Manton construction of Skyrmions from instanton holonomy.