  We study traveling wave solutions for a nonlinear Schr\"odinger system with
quadratic interaction. For the non mass resonance case, the system has no
Galilean symmetry, which is of particular interest in this paper. We construct
traveling wave solutions by variational methods and see that for the non mass
resonance case there exist specific traveling wave solutions which correspond
to the solutions for ``zero mass" case in nonlinear elliptic equations. We also
establish the new global existence result for oscillating data as an
application. Both of our results essentially come from the lack of Galilean
invariance in the system.
