Dispersive blow-up for a coupled Schrödinger-fifth order KdV system
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| Format: | Preprint |
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2024
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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 establish a dispersive blow-up result for the initial value problem (IVP) for the coupled Schrödinger-fifth order Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=αuv + γ|u|^2 u, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ \partial_t v + \partial_x^5 v + \partial_x v^2&\hspace{-2mm}=ε\partial_x |u|^2, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ u(x,0)&\hspace{-2mm}= u_0(x), \quad v(x,0)=v_0(x). \end{array} \right\} \end{align*} To achieve this, we prove a local well-posedness result in Bourgain spaces of the type $X^{s+β,b}\times Y^{s,b}$, along with a regularity property for the nonlinear part of the IVP solutions. This property enables the construction of initial data that leads to the dispersive blow-up phenomenon. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08759 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dispersive blow-up for a coupled Schrödinger-fifth order KdV system Bustamante, Eddye Urrea, José Jiménez Mejía, Jorge Analysis of PDEs In this work we establish a dispersive blow-up result for the initial value problem (IVP) for the coupled Schrödinger-fifth order Korteweg-de Vries system \begin{align*} \left. \begin{array}{rl} i u_t+\partial_x^2 u &\hspace{-2mm}=αuv + γ|u|^2 u, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ \partial_t v + \partial_x^5 v + \partial_x v^2&\hspace{-2mm}=ε\partial_x |u|^2, \quad x\in\mathbb R,\quad t\in\mathbb R,\\ u(x,0)&\hspace{-2mm}= u_0(x), \quad v(x,0)=v_0(x). \end{array} \right\} \end{align*} To achieve this, we prove a local well-posedness result in Bourgain spaces of the type $X^{s+β,b}\times Y^{s,b}$, along with a regularity property for the nonlinear part of the IVP solutions. This property enables the construction of initial data that leads to the dispersive blow-up phenomenon. |
| title | Dispersive blow-up for a coupled Schrödinger-fifth order KdV system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2412.08759 |