  We consider the Cauchy problem of the system of nonlinear Schr\"odinger
equations with derivative nonlinearlity. This system was introduced by
Colin-Colin (2004) as a model of laser-plasma interactions. We study existence
of ground state solutions and the global well-posedness of this system by using
the variational methods. We also consider the stability of traveling waves for
this system. These problems are proposed by Colin-Colin as the open problems.
We give a subset of the ground-states set which satisfies the condition of
stability. In particular, we prove the stability of the set of traveling waves
with small speed for $1$-dimension.
