  We study pattern-forming nonlinear dynamics starting from a continuous wave
state of quasi-one-dimensional two-component Bose-Einstein condensates with
synthetic spin-orbit coupling induced by Raman lasers. Modulation instability
can occur even when the miscibility condition due to the interatomic
interactions is satisfied. We find that the initial stage of the nonlinear
development is consistent with the prediction of modulation instability, where
the two primary and secondary instability bands lead to the spontaneous growth
of the modulation and the subsequent complicated dynamics of pattern formation.
At later stages of the evolution, the wave functions undergo clear separation
in the momentum space, reflected in the dispersion of the single particle
Hamiltonian.
