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We demonstrate the formation of stable spatially-ordered states in a {\ituniform} and also {\it trapped} quasi-two-dimensional (quasi-2D) Rashba orDresselhaus spin-orbit (SO) coupled pseudo spin-1/2 Bose-Einstein condensateusing the mean-field Gross-Pitaevskii equation. For weak SO coupling, one canhave a circularly-symmetric $(0,+1)$- or $(0,-1)$-type multi-ring state withintrinsic vorticity, for Rashba or Dresselhaus SO coupling, respectively, wherethe numbers in the parentheses denote the net angular momentum projection inthe two components, in addition to a circularly-asymmetric degenerate statewith zero net angular momentum projection. For intermediate SO couplings, inaddition to the above two types, one can also have states with stripe patternin component densities with no periodic modulation in total density. The stripestate continues to exist for large SO coupling. In addition, a newspatially-periodic state appears in the uniform system: a superlattice state,possessing some properties of a supersolid, with a square-lattice pattern incomponent densities and also in total density. In a trapped system thesuperlattice state is slightly different with multi-ring pattern in componentdensity and a square-lattice pattern in total density. For an equal mixture ofRashba and Dresselhaus SO couplings, in both uniform and trapped systems, onlystripe states are found for all strengths of SO couplings. In a uniform systemall these states are quasi-2D solitonic states.

Spatial order in a two-dimensional spin-orbit-coupled spin-1/2 condensate: superlattice, multi-ring and stripe formation