  It is known that stable 2D solitons of the semi-vortex (SV) and mixed-mode
(MM) types are maintained by the interplay of the cubic attractive nonlinearity
and spin-orbit coupling (SOC) in binary Bose-Einstein condensates. We introduce
a double-layer system, in which two binary condensates, stabilized by the SOC,
are linearly coupled by tunneling. By means of the numerical methods, it is
found that symmetric two-layer solitons undergo the
spontaneous-symmetry-breaking (SSB) bifurcation of the subcritical type. The
bifurcation produces families of asymmetric 2D solitons, which exist up to the
value of the total norm equal to the norm of the Townes solitons, above which
the collapse occurs. This situation terminates at a critical value of the
inter-layer coupling, beyond which the SSB bifurcation is absent, as the
collapse sets in earlier. Symmetric 2D solitons that are destabilized by the
SSB demonstrate dynamical symmetry breaking, in combination with intrinsic
oscillations of the solitons, or transition to the collapse, if the soliton's
norm is sufficiently large. Asymmetric MMs produced by the SSB instability
start spontaneous drift, in addition to the intrinsic vibrations. Consideration
of moving 2D solitons is a nontrivial problem because SOC breaks the Galilean
invariance. It is found that the system supports moving MMs up to a critical
value of the velocity, beyond which they undergo delocalization.

Quantum Gases