Spontaneous spatial order in two-dimensional ferromagnetic spin-orbit coupled uniform spin-1 condensate solitons

We demonstrate spontaneous spatial order in stripe and super-lattice solitons in a Rashba spin-orbit (SO) coupled spin-1 uniform quasi-two-dimensional (quasi-2D) ferromagnetic Bose-Einstein condensate. For weak SO coupling, the solitons are of the $(- 1,0,+ 1)$ or $(0,+ 1,+ 2)$ type with intrinsic vorticity, where the numbers in the parentheses denote angular momentum in spin components $F_z=+1,0,-1$, respectively. For intermediate SO coupling, three types of solitons are found: (a) circularly-asymmetric solitons, (b) circularly-symmetric $(- 1,0,+ 1)$- and (c) $(0,+ 1,+ 2)$-type multi-ring solitons maintaining the above-mentioned vortices in respective components. For large SO coupling, quasi-degenerate stripe and super-lattice solitons are found in addition to the circularly-asymmetric solitons. A super-lattice soliton forms a 2D square lattice structure in the total density as in a super-solid; in component densities it may have either (i) a 1D stripe pattern or (ii) a 2D square lattice structure.