Solitary waves (solitons, standing waves, traveling waves) are known to play a fundamental role in the analysis of nonlinear dispersive equations. First, we briefly explain nonlinear dispersive equations and solitary waves by using the nonlinear Schrödinger equation as an example. Next, as an introduction of recent results obtained by the speaker (jointly with N. Fukaya and T. Inui), we consider traveling waves of a nonlinear Schrödinger sys- tem with quadratic interaction. When the mass constants satisfy a certain condition, the system has no specific symmetry such as Galilean or pseudo-conformal symmetry, which is of particular interest. We construct traveling wave solutions by variational methods, and also show global existence for oscillating large data as an application. We see that our results essentially come from the lack of Galilean invariance in the system. Finally, I will introduce a few open problems and interesting relevance to other types of nonlinear dispersive equations.