  In the present work, we explore the potential of spin-orbit (SO) coupled
Bose-Einstein condensates to support multi-component solitonic states in the
form of dark-bright (DB) solitons. In the case where Raman linear coupling
between components is absent, we use a multiscale expansion method to reduce
the model to the integrable Mel'nikov system. The soliton solutions of the
latter allow us to reconstruct approximate traveling DB solitons for the
reduced SO coupled system. For small values of the formal perturbation
parameter, the resulting waveforms propagate undistorted, while for large
values thereof, they shed some dispersive radiation, and subsequently distill
into a robust propagating structure. After quantifying the relevant radiation
effect, we also study the dynamics of DB solitons in a parabolic trap,
exploring how their oscillation frequency varies as a function of the bright
component mass and the Raman laser wavenumber.
