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We consider the Cauchy problem of the system of nonlinear Schr\"odingerequations with derivative nonlinearlity. This system was introduced byColin-Colin (2004) as a model of laser-plasma interactions. We study existenceof ground state solutions and the global well-posedness of this system by usingthe variational methods. We also consider the stability of traveling waves forthis 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 ofstability. In particular, we prove the stability of the set of traveling waveswith small speed for $1$-dimension.

Variational problems for the system of nonlinear Schr\"odinger equations with derivative nonlinearities