On long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions
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| Format: | Preprint |
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2025
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| _version_ | 1866912289113767936 |
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| author | Linares, Felipe Zhou, Dequin |
| author_facet | Linares, Felipe Zhou, Dequin |
| contents | We consider the long time behavior of the solutions of the coupled Schrödinger-KdV systems \begin{eqnarray*} \left\{ \begin{array}{llll}i\partial_tu+\partial^2_xu=αuv+βu|u|^2,\hskip30pt (x,t)\in \mathbb{R}\times \mathbb{R}^{+},\\ \partial_tv+\partial^3_xv+v\partial_xv=γ\partial_x(|u|^2), \hskip20pt (x,t)\in \mathbb{R}\times \mathbb{R}^{+},\\ u, v)|_{t=0} =(u_{0}, v_{0}). \end{array} \right. \end{eqnarray*} We show that global solutions to this system satisfy locally energy decay in a suitable interval, growing unbounded in time, in two situations. In the first case, we regard the parameter vector $(α,β,γ)\in \mathbb{R}^{+}\times \overline{\mathbb{R}^{+}}\times \mathbb{R}^{+}$ without any size assumption on the initial data in $ H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. In the second one, we consider the parameter vector $(α,β,γ)\in \mathbb{R}^{+}\times \mathbb{R}^{-}\times \mathbb{R}^{+}$. In this case, we give a \lq\lq smallness" criterion involving the product of the parameter $-β$ and a constant depending on the initial data in $H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. Our results answer positively the open questions raised in [F. Linares, A. J. Mendez, SIAM J. Math. Anal. 53(2021) 3838-3855]. We use new ideas and different techniques from the latter paper. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_17775 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions Linares, Felipe Zhou, Dequin Analysis of PDEs 35Q53, 35B35, 35B40, 35Q55 We consider the long time behavior of the solutions of the coupled Schrödinger-KdV systems \begin{eqnarray*} \left\{ \begin{array}{llll}i\partial_tu+\partial^2_xu=αuv+βu|u|^2,\hskip30pt (x,t)\in \mathbb{R}\times \mathbb{R}^{+},\\ \partial_tv+\partial^3_xv+v\partial_xv=γ\partial_x(|u|^2), \hskip20pt (x,t)\in \mathbb{R}\times \mathbb{R}^{+},\\ u, v)|_{t=0} =(u_{0}, v_{0}). \end{array} \right. \end{eqnarray*} We show that global solutions to this system satisfy locally energy decay in a suitable interval, growing unbounded in time, in two situations. In the first case, we regard the parameter vector $(α,β,γ)\in \mathbb{R}^{+}\times \overline{\mathbb{R}^{+}}\times \mathbb{R}^{+}$ without any size assumption on the initial data in $ H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. In the second one, we consider the parameter vector $(α,β,γ)\in \mathbb{R}^{+}\times \mathbb{R}^{-}\times \mathbb{R}^{+}$. In this case, we give a \lq\lq smallness" criterion involving the product of the parameter $-β$ and a constant depending on the initial data in $H^{1}(\mathbb{R})\times H^{1}(\mathbb{R})$. Our results answer positively the open questions raised in [F. Linares, A. J. Mendez, SIAM J. Math. Anal. 53(2021) 3838-3855]. We use new ideas and different techniques from the latter paper. |
| title | On long time behavior of solutions of the Schrödinger-KdV system with and without resonant interactions |
| topic | Analysis of PDEs 35Q53, 35B35, 35B40, 35Q55 |
| url | https://arxiv.org/abs/2503.17775 |