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The dynamics of non-autonomous bright matter-wave solitons in Rabi coupledbinary Bose-Einstein condensates is explored. By performing a unitary andsimilarity/lens-type transformation, we reduce the non-autonomousGross-Pitaevskii equation into the celebrated Manakov model. Then, we constructthe exact bright solitons of the non-autonomous Gross-Pitaevskii system in thepresence of time dependent nonlinearities for two specific forms, namelyhyperbolic nonlinearities, which are of physical interest. The experimentalpossibilities of realizing the forms of temporally modulated potentialscorresponding to these time dependent nonlinearities and supporting suchlocalized structures are explored. Our study on the propagation of one solitonshows that the amplitude, velocity and shape of the bright soliton are alteredby the time-dependent scattering length. We also analyse the non-trivial energysharing collision of Manakov solitons in the presence of Rabi coupling andaforementioned nonlinearities. We find that breathers arise in two-solitoncollisions and the nature of energy sharing collisions is altered from that ofManakov system due to Rabi coupling only. Further in the presence oftime-dependent nonlinearities the collision scenario is again alteredsignificantly. Finally, the stability of these localized structures is examinedusing a recently developed powerful analytic method by Quintero et. al., [Phys.Rev. E 91, 012905 (2015)] and it is shown that the non- autonomous brightsolitons are indeed stable. The evolution of position and velocity is alsostudied.

Non-autonomous bright solitons and their stability in Rabi coupled binary Bose-Einstein Condensates