Dark soliton-like magnetic domain walls in a two-dimensional ferromagnetic superfluid

We report a stable magnetic domain wall in a uniform ferromagnetic spin-1 condensate, characterized by the magnetization having a dark soliton profile with nonvanishing superfluid density. We find exact stationary solutions for a particular ratio of interaction parameters with and without magnetic fields, and develop an accurate analytic solution applicable to the whole ferromagnetic phase. In the absence of magnetic fields, this domain wall relates various distinct solitary excitations in binary condensates through $\textrm{SO}(3)$ spin rotations, which otherwise are unconnected. Remarkably, studying the dynamics of a quasi-two-dimensional (quasi-2D) system we show that standing wave excitations of the domain wall oscillate without decay, being stable against the snake instability. The domain wall is dynamically unstable to modes that cause the magnetization to grow perpendicularly while leaving the domain wall unchanged. Real time dynamics in the presence of white noise reveals that this "spin twist" instability does not destroy the topological structure of the magnetic domain wall.