A note on the uniqueness properties of solutions for the Schrödinger-Korteweg de Vries system
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912461200818176 |
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| author | Bustamante, Eddye Urrea, José Jiménez Mejía, Jorge |
| author_facet | Bustamante, Eddye Urrea, José Jiménez Mejía, Jorge |
| contents | In this work we prove that if $(u_i,v_i)$, $i=1,2$, are smooth enough solutions of the coupled Schrödinger-Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=βuv - |u|^2 u,\\ \partial_t v + \partial_x^3 v &\hspace{-2mm}=γ\partial_x |u|^2-\frac12\partial_x (v^2) \end{array} \right\} \end{align*} with appropriate decay at infinity such that at two different times $t_0=0$ and $t_1=1$ satisfy that $$u_1(0)-u_2(0),u_1(1)-u_2(1),v_1(0)-v_2(0),v_1(1)-v_2(1)\in H^1(e^{ax^{2}}dx),$$ for $a>0$ big enough, then $u_1=u_2$ and $v_1=v_2$.
(Let us recall that $f\in H^1(e^{ax^{2}} dx)$ iff $f\in L^2(e^{ax^{2}}dx)$ and $\partial_x f\in L^2(e^{ax^{2}}dx)$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01733 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the uniqueness properties of solutions for the Schrödinger-Korteweg de Vries system Bustamante, Eddye Urrea, José Jiménez Mejía, Jorge Analysis of PDEs In this work we prove that if $(u_i,v_i)$, $i=1,2$, are smooth enough solutions of the coupled Schrödinger-Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=βuv - |u|^2 u,\\ \partial_t v + \partial_x^3 v &\hspace{-2mm}=γ\partial_x |u|^2-\frac12\partial_x (v^2) \end{array} \right\} \end{align*} with appropriate decay at infinity such that at two different times $t_0=0$ and $t_1=1$ satisfy that $$u_1(0)-u_2(0),u_1(1)-u_2(1),v_1(0)-v_2(0),v_1(1)-v_2(1)\in H^1(e^{ax^{2}}dx),$$ for $a>0$ big enough, then $u_1=u_2$ and $v_1=v_2$. (Let us recall that $f\in H^1(e^{ax^{2}} dx)$ iff $f\in L^2(e^{ax^{2}}dx)$ and $\partial_x f\in L^2(e^{ax^{2}}dx)$). |
| title | A note on the uniqueness properties of solutions for the Schrödinger-Korteweg de Vries system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.01733 |