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Josephson oscillations of chirality and identity in two-dimensional solitons in spin-orbit-coupled condensates

We investigate dynamics of two-dimensional chiral solitons of semi-vortex (SV) and mixed-mode (MM) types in spin-orbit-coupled Bose-Einstein condensates with the Manakov nonlinearity, loaded in a dual-core (double-layer) trap. The system supports two novel manifestations of Josephson phenomenology: one in the form of persistent oscillations between SVs or MMs with opposite chiralities in the two cores, and another one demonstrating robust periodic switching (identity oscillations) between SV in one core and MM in the other, provided that the strength of the inter-core coupling exceeds a threshold value. Below the threshold, the system creates composite states, which are asymmetric with respect to the two cores, or suffer the collapse. Robustness of the chirality and identity oscillations against deviations from the Manakov nonlinearity is investigated too. These dynamical regimes are possible only in the nonlinear system. In the linear one, exact stationary and dynamical solutions for SVs and MMs of the Bessel type are found. They sustain Josephson self-oscillations in different modes, with no interconversion between them.