On solitary wave solutions with two-frequency parameters to the three-component system of quadratic nonlinear Schr\"odinger equations

In the present paper, we consider the Cauchy problem of a system of three nonlinear Schr"odinger equations with quadratic nonlinearity. We first prove the existence of ground states in the form of solitary wave solutions with two frequency parameters. We then show that the conditions on the frequency parameters for the existence of ground states depend on the resonance structure of the system. Next, we give the two results for global solutions. The first is an improvement of global well-posedness for initial data below the ground state threshold. The second is an improvement of global well-posedness for oscillating initial data. We also prove the orbital stability of the ground state sets with small speed parameter.