J-kink domain walls and the DBI action

We study $J$-kink domain walls in $D=4$ massive $\mathbb{C}P^1$ sigma model. The domain walls are not static but stationary, since they rotate in an internal $S^1$ space with a frequency $\omega$ and a momentum ${\bf k}$ along the domain wall. They are characterized by a conserved current $J_\mu = (Q,{\bf J})$, and are classified into magnetic ($J^2 < 0$), null ($J^2=0$), and electric ($J^2 > 0$) types. Under a natural assumption that a low energy effective action of the domain wall is dual to the $D=4$ DBI action for a membrane, we are lead to a coincidence between the $J$-kink domain wall and the membrane with constant magnetic field $B$ and electric field ${\bf E}$. We also find that $(Q, {\bf J}, \omega, {\bf k})$ is dual to $(B, {\bf E}, H, {\bf D})$ with $H$ and ${\bf D}$ being a magnetizing field and a displacement field, respectively.