Traveling waves for a nonlinear Schr\"odinger system with quadratic interaction

TL;DR

This paper investigates traveling wave solutions for a nonlinear Schrödinger system with quadratic interaction, focusing on the non mass resonance case where Galilean symmetry is absent, using variational methods and establishing new global existence results.

Contribution

It constructs specific traveling wave solutions in the non mass resonance case and proves a new global existence result for oscillating data, highlighting effects of the lack of Galilean invariance.

Findings

Existence of traveling wave solutions in non mass resonance case

Construction of solutions via variational methods

New global existence result for oscillating initial data

Abstract

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.