Spontaneous symmetry and antisymmetry breaking of two-component solitons in a combination of linear and nonlinear double-well potentials

We investigate the phenomenology of spontaneous symmetry breaking (SSB) and spontaneous antisymmetry breaking (SASB) in the framework of a system of coupled nonlinear Schrödinger equations (NLSEs) with a combination of linear and nonlinear double-well potentials (DWPs), including self- and cross-phase-modulation (SPM and XPM) nonlinear terms. The DWP structure is represented by a symmetric pair of delta functions. This setup can be realized in optics and for matter waves in Bose-Einstein condensates. It is commonly known that SSB states are completely stable in the single one-dimensional NLSE with a combination of DWP and attractive SPM nonlinearity. The objective of the present work is to elaborate on the SSB and SASB phenomenology in the two-component system. One factor that affects the phenomenology (actually, the stability of stationary states) in this context is the relative sign of the two components, the states with identical/opposite signs defined as ones with in-phase/out-of-phase components. We demonstrate that the SSB/SASB, with both components originating from spatially symmetric or antisymmetric modes, of the out-of-phase type, in the coupled NLSEs including the SPM and XPM terms with opposite signs, are similar to their counterparts in the single NLSE. On the other hand, the SSB/SASB of in-phase states in the coupled NLSEs, including the SPM and XPM terms with opposite signs, are quite different from the ones in the single NLSE. In particular, we find that the in-phase SSB/SASB states are unstable in a part of the intervals of their propagation constant if the system includes SPM and XPM terms with opposite signs. As another feature that is possible only in the two-component system, we address mixed-parity states, originally built of spatially symmetric and antisymmetric components, and find that such states, featuring SSB or SASB, are stable in a part of their interval of the propagation constant. We clarify that the spatially antisymmetric component is a dominant one in determining the asymmetry of the established state if it is coupled to the symmetric one, in the case of the repulsive SPM. On the other hand, the symmetric component dominates if it is coupled to the antisymmetric one in the case of the attractive SPM.