We study a nonlinear Schrödinger system with three-wave interaction: \(\begin{aligned} \left\{ \begin{aligned}&- \Delta u_1 = f_1(u_1) + \alpha u_2u_3 \quad \text { in } \mathbb {R}^N, \\&- \Delta u_2 = f_2(u_2) + \alpha u_3u_1 \quad \text { in } \mathbb {R}^N, \\&- \Delta u_3 = f_3(u_3) + \alpha u_1u_2 \quad \text { in } \mathbb {R}^N, \\&\quad \vec {u}=(u_1,u_2,u_3)\in (H_\textrm{rad}^1(\mathbb {R}^N))^3, \end{aligned}\right. \end{aligned}\) where \(3\le N\le 5\) , \(\alpha \in \mathbb {R}\) 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 \(H_\textrm{rad}^1(\mathbb {R}^N)\) . 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 \(\alpha \rightarrow 0\) . One family satisfies \(\textrm{dist}(\vec {u}_{\alpha },S_1\times S_2\times S_3) \rightarrow 0\) , while another satisfies \(\textrm{dist}(\vec {u}_{\alpha },S_1\times S_2\times \{0\}) \rightarrow 0\) . By contrast, we prove that no family of vector solutions satisfies \(\textrm{dist}(\vec {u}_{\alpha },S_1\times \{0\}\times \{0\}) \rightarrow 0\) . Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak.
