On the existence of vector solutions to nonlinear Schr\"odinger equations with weak three-wave interaction

TL;DR

This paper investigates the existence and asymptotic behavior of vector solutions to a nonlinear Schrödinger system with three-wave interaction, revealing two distinct families of solutions as the interaction weakens.

Contribution

It establishes the existence of two families of vector solutions with different asymptotic limits and shows the non-existence of solutions approaching certain trivial configurations.

Findings

Two families of vector solutions with different limits as interaction strength goes to zero.

One family approaches solutions with all components nontrivial, another approaches solutions with two components nontrivial.

No solutions approach configurations with only one nontrivial component.

Abstract

We study a nonlinear Schr\"odinger system with three-wave interaction: \begin{equation*} \left\{\begin{aligned} & - \Delta u_1 = f_1(u_1) + \alpha u_2u_3 \quad \text{ in } \R^N, & - \Delta u_2 = f_2(u_2) + \alpha u_3u_1 \quad \text{ in } \R^N, & - \Delta u_3 = f_3(u_3) + \alpha u_1u_2 \quad \text{ in } \R^N, & \quad \vec{u}=(u_1,u_2,u_3)\in (H_{\rm rad}^1(\R^N))^3, \end{aligned}\right. \end{equation*} where $3\leq N\leq 5$, $\alpha\in \R$ and each nonlinearity $f_i(\xi)$ satisfies the Berestycki-Lions conditions. Let $S_i$ denote the set of all least energy solutions of the scalar equation $-\Delta u = f_i(u)$ in $H_{\rm rad}^1(\R^N)$. A solution of the systems is called vector if all its components are nontrivial. We establish the existence of two distinct families of vector solutions $\{\vec{u}_\alpha\}$ with different asymptotic behaviors as $\alpha \to 0$. One family satisfies ${\rm dist}(\vec{u}_{\alpha},S_1\times S_2\times S_3) \to 0$, while another satisfies ${\rm dist}(\vec{u}_{\alpha},S_1\times S_2\times \{0\}) \to 0$. By contrast, we prove that no family of vector solutions satisfies ${\rm dist}(\vec{u}_{\alpha},S_1\times \{0\}\times \{0\}) \to 0$. Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak.