Traveling waves of NLS System arising in optical material without Galilean symmetry

Abstract

We consider a system of NLS with cubic interactions arising in nonlinear optics without Galilean symmetry. The absence of Galilean symmetry can lead to many difficulties, such as global existence and blowup problems; see [Comm. Partial Differential Equations 46, 11 (2021), 2134-2170]. In this paper, we mainly focus on the influence of the absence of this symmetry on the traveling waves of the NLS system. Firstly, we obtain the existence of traveling solitary wave solutions that are non-radial and complex-valued. Secondly, using the asymptotic analysis method, when the frequency is sufficiently large, we establish the high frequency limit of the traveling solitary wave solution. Finally, for the mass critical case, we provide a novel condition for the existence of global solutions which is significantly different from the classical. In particular, this new condition breaks the traditional optimal assumption about initial data.