  We investigate the dynamics of a spin-orbit (SO) coupled BECs in a time
dependent harmonic trap and show the dynamical system to be completely
integrable by constructing the Lax pair. We then employ gauge transformation
approach to witness the rapid oscillations of the condensates for a relatively
smaller value of SO coupling in a time independent harmonic trap compared to
their counterparts in a transient trap. Keeping track of the evolution of the
condensates in a transient trap during its transition from confining to
expulsive trap, we notice that they collapse in the expulsive trap. We further
show that one can manipulate the scattering length through Feshbach resonance
to stretch the lifetime of the confining trap and revive the condensate.
Considering a SO coupled state as the initial state, the numerical simulation
indicates that the reinforcement of Rabi coupling on SO coupled BECs generates
the striped phase of the bright solitons and does not impact the stability of
the condensates despite destroying the integrability of the dynamical system.
