  The dynamics of non-autonomous bright matter-wave solitons in Rabi coupled
binary Bose-Einstein condensates is explored. By performing a unitary and
similarity/lens-type transformation, we reduce the non-autonomous
Gross-Pitaevskii equation into the celebrated Manakov model. Then, we construct
the exact bright solitons of the non-autonomous Gross-Pitaevskii system in the
presence of time dependent nonlinearities for two specific forms, namely
hyperbolic nonlinearities, which are of physical interest. The experimental
possibilities of realizing the forms of temporally modulated potentials
corresponding to these time dependent nonlinearities and supporting such
localized structures are explored. Our study on the propagation of one soliton
shows that the amplitude, velocity and shape of the bright soliton are altered
by the time-dependent scattering length. We also analyse the non-trivial energy
sharing collision of Manakov solitons in the presence of Rabi coupling and
aforementioned nonlinearities. We find that breathers arise in two-soliton
collisions and the nature of energy sharing collisions is altered from that of
Manakov system due to Rabi coupling only. Further in the presence of
time-dependent nonlinearities the collision scenario is again altered
significantly. Finally, the stability of these localized structures is examined
using a recently developed powerful analytic method by Quintero et. al., [Phys.
Rev. E 91, 012905 (2015)] and it is shown that the non- autonomous bright
solitons are indeed stable. The evolution of position and velocity is also
studied.
