  We consider a quasi-one-dimensional two-component Bose-Einstein condensate
subject to a coherent coupling between its components, such as realized in
spin-orbit coupled condensates. We study how nonlinearity modifies the dynamics
of the elementary excitations. The spectrum has two branches which are affected
in different ways. The upper branch experiences a modulational instability
which is stabilized by a long wave-short wave resonance with the lower branch.
The lower branch is stable. In the limit of weak nonlinearity and small
dispersion it is described by a Korteweg-de Vries equation or by the Gardner
equation, depending on the value of the parameters of the system.
