On the existence of vector solutions to nonlinear Schrödinger equations with weak three-wave interaction
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910111990022144 |
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| author | Kinoshita, T. Sato, Y. |
| author_facet | Kinoshita, T. Sato, Y. |
| contents | We study a nonlinear Schrödinger system with three-wave interaction: \begin{equation*}
\left\{\begin{aligned}
& - Δu_1 = f_1(u_1) + αu_2u_3 \quad \text{ in } \R^N,
& - Δu_2 = f_2(u_2) + αu_3u_1 \quad \text{ in } \R^N,
& - Δu_3 = f_3(u_3) + αu_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 $3\leq N\leq 5$, $α\in \R$ and each nonlinearity $f_i(ξ)$ satisfies the Berestycki-Lions conditions. Let $S_i$ denote the set of all least energy solutions of the scalar equation $-Δu = f_i(u)$ 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 $\{\vec{u}_α\}$ with different asymptotic behaviors as $α\to 0$. One family satisfies ${\rm dist}(\vec{u}_α,S_1\times S_2\times S_3) \to 0$, while another satisfies ${\rm dist}(\vec{u}_α,S_1\times S_2\times \{0\}) \to 0$. By contrast, we prove that no family of vector solutions satisfies ${\rm dist}(\vec{u}_α,S_1\times \{0\}\times \{0\}) \to 0$. Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06678 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence of vector solutions to nonlinear Schrödinger equations with weak three-wave interaction Kinoshita, T. Sato, Y. Analysis of PDEs We study a nonlinear Schrödinger system with three-wave interaction: \begin{equation*} \left\{\begin{aligned} & - Δu_1 = f_1(u_1) + αu_2u_3 \quad \text{ in } \R^N, & - Δu_2 = f_2(u_2) + αu_3u_1 \quad \text{ in } \R^N, & - Δu_3 = f_3(u_3) + αu_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 $3\leq N\leq 5$, $α\in \R$ and each nonlinearity $f_i(ξ)$ satisfies the Berestycki-Lions conditions. Let $S_i$ denote the set of all least energy solutions of the scalar equation $-Δu = f_i(u)$ 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 $\{\vec{u}_α\}$ with different asymptotic behaviors as $α\to 0$. One family satisfies ${\rm dist}(\vec{u}_α,S_1\times S_2\times S_3) \to 0$, while another satisfies ${\rm dist}(\vec{u}_α,S_1\times S_2\times \{0\}) \to 0$. By contrast, we prove that no family of vector solutions satisfies ${\rm dist}(\vec{u}_α,S_1\times \{0\}\times \{0\}) \to 0$. Together, these results give a complete description of the asymptotic structure of vector solutions when the three-wave interaction is weak. |
| title | On the existence of vector solutions to nonlinear Schrödinger equations with weak three-wave interaction |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.06678 |