Global well-posedness and orbital stability of solitary waves for Zakharov-Ito equation
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| Format: | Preprint |
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2025
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| _version_ | 1866915345664573440 |
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| author | Wu, Fan Shao, Feng |
| author_facet | Wu, Fan Shao, Feng |
| contents | In this paper, we consider the Zakharov-Ito equation \begin{equation*} \begin{cases} u_t+u_{xxx}+3uu_x+ρρ_x=0,\\ ρ_t+{(uρ)}_x=0. \end{cases} \end{equation*} We prove the local well-posedness in $H^s\times H^s$ for $s>3/2$ and global well-posedness in $H^s\times H^s$ for $s\geq2$. When $ρ=0$, the Zakharov-Ito equation reduces to the KdV equation, hence has solitary waves with speeds $c\in(0,+\infty)$. We prove the orbital stability of these solitary waves in $H^1\times L^2$ by combining a variational approach and the framework of Grillakis, Shatah and Strauss \cite{GSS1987}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07053 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global well-posedness and orbital stability of solitary waves for Zakharov-Ito equation Wu, Fan Shao, Feng Analysis of PDEs Mathematical Physics In this paper, we consider the Zakharov-Ito equation \begin{equation*} \begin{cases} u_t+u_{xxx}+3uu_x+ρρ_x=0,\\ ρ_t+{(uρ)}_x=0. \end{cases} \end{equation*} We prove the local well-posedness in $H^s\times H^s$ for $s>3/2$ and global well-posedness in $H^s\times H^s$ for $s\geq2$. When $ρ=0$, the Zakharov-Ito equation reduces to the KdV equation, hence has solitary waves with speeds $c\in(0,+\infty)$. We prove the orbital stability of these solitary waves in $H^1\times L^2$ by combining a variational approach and the framework of Grillakis, Shatah and Strauss \cite{GSS1987}. |
| title | Global well-posedness and orbital stability of solitary waves for Zakharov-Ito equation |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2506.07053 |