Traveling waves for a nonlinear Schrödinger system with quadratic interaction

TL;DR

The paper investigates traveling wave solutions for a two-component nonlinear Schrödinger system with quadratic interaction in the non mass resonance setting ($\kappa\neq \tfrac12$), where Galilean invariance is lost. ItDevelops a two-parameter variational framework using the action $S_{\omega,c}$ and Nehari manifold to construct traveling waves of the form $(u,v)=(e^{i\omega t}\phi_{\omega,c}(x-ct), e^{i2\omega t}\psi_{\omega,c}(x-ct))$, establishing existence in several regimes including a zero-mass scenario. It also proves new global existence results in the $L^2$-critical case with oscillating data in $d=4$ by potential-well methods, showing how large momentum boosts can enforce global behavior when $\kappa\neq \tfrac12$. Altogether, the work highlights how the absence of Galilean invariance yields novel nonradial traveling waves and modifies global dynamics, contributing to the understanding of nonresonant NLS systems and their long-time behavior.

Abstract

We study traveling wave solutions for a nonlinear Schrödinger 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.

Theorems & Definitions (36)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• proof
• Lemma 2.4