Traveling Waves for a System of NLS Arising in Optical Materials With χ(3) Nonlinear Response

In this paper, we investigate the dynamics of traveling wave solutions for a class of nonlinear Schödinger equations with nonlinear response under the non mass resonance case. We are interested in the non mass resonance case where the system does not have Galilean symmetry. First, using the variational methods and the concentration‐compactness arguments, we prove the existence of traveling wave solutions in the “zero mass” case. Then, we obtain the global well‐posedness in the ‐subcritical case by using Gagliardo–Nirenberg (G–N) type inequality. In particular, we also obtain the global well‐posedness in the ‐critical case and ‐supercritical case respectively, by finding the best constant of the corresponding Gagliardo–Nirenberg (G–N) type inequality. In addition, we establish the blow‐up result corresponding of traveling wave solutions based on localized Virial estimates, and build up a new blow‐up criterion.