  We investigate 1D and 2D radial domain-wall (DW) states in the system of two
nonlinear-Schr\"{o}dinger/Gross-Pitaevskii equations, which are coupled by the
linear mixing and by the nonlinear XPM (cross-phase-modulation). The system has
straightforward applications to two-component Bose-Einstein condensates, and to
the bimodal light propagation in nonlinear optics. In the former case, the two
components represent different hyperfine atomic states, while in the latter
setting they correspond to orthogonal polarizations of light. Conditions
guaranteeing the stability of flat continuous wave (CW) asymmetric bimodal
states are established, followed by the study of families of the corresponding
DW patterns. Approximate analytical solutions for the DWs are found near the
point of the symmetry-breaking bifurcation of the CW states. An exact DW
solution is produced for ratio 3:1 of the XPM and SPM coefficients. The DWs
between flat asymmetric states, which are mirror images to each other, are
completely stable, and all other species of the DWs, with zero crossings in one
or two components, are fully unstable. Interactions between two DWs are
considered too, and an effective potential accounting for the attraction
between them is derived analytically. Direct simulations demonstrate merger and
annihilation of the interacting DWs. The analysis is extended for the system
including single- and double-peak external potentials. Generic solutions for
trapped DWs are obtained in a numerical form, and their stability is
investigated. An exact stable solution is found for the DW trapped by a
single-peak potential. In the 2D geometry, stable two-component vortices are
found, with topological charges s=1,2,3. Radial oscillations of annular
DW-shaped pulsons, with s=0,1,2, are studied too. A linear relation between the
period of the oscillations and the mean radius of the DW ring is derived
analytically.
