  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.
