On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach
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| Format: | Preprint |
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2025
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| author | Boukarou, Aissa Seghour, Lamia |
| author_facet | Boukarou, Aissa Seghour, Lamia |
| contents | In this paper, we improve and extend the results obtained by Boukarou et al. \cite{boukarou1} on the Gevrey regularity of solutions to a fifth-order Kadomtsev-Petviashvili-II equation. We establish Gevrey regularity in the time variable for solutions in $2+1$ dimensions, providing a sharper result obtained through a new analytical approach. Assuming that the initial data are Gevrey regular of order $σ\geq 1$ in the spatial variables, we prove that the corresponding solution is Gevrey regular of order $5 σ$ in time. Moreover, we show that the function $u(x, y, t)$, viewed as a function of $t$, does not belong to $G^z$ for any $1 \leq z<5 σ$. Our proof introduces a new analytical method that establishes a general principle for dispersive equations of the form $ \partial_t u = \pm\partial_x^αu + P(u),$ where $\partial_x^α$ is the highest spatial derivative and $P(u)$ a polynomial in spatial derivatives of total order at most $α-1$, the solution cannot belong to the Gevrey class $G^z$ in time for any $z$ satisfying $1 \leq z<ασ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26669 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach Boukarou, Aissa Seghour, Lamia Analysis of PDEs In this paper, we improve and extend the results obtained by Boukarou et al. \cite{boukarou1} on the Gevrey regularity of solutions to a fifth-order Kadomtsev-Petviashvili-II equation. We establish Gevrey regularity in the time variable for solutions in $2+1$ dimensions, providing a sharper result obtained through a new analytical approach. Assuming that the initial data are Gevrey regular of order $σ\geq 1$ in the spatial variables, we prove that the corresponding solution is Gevrey regular of order $5 σ$ in time. Moreover, we show that the function $u(x, y, t)$, viewed as a function of $t$, does not belong to $G^z$ for any $1 \leq z<5 σ$. Our proof introduces a new analytical method that establishes a general principle for dispersive equations of the form $ \partial_t u = \pm\partial_x^αu + P(u),$ where $\partial_x^α$ is the highest spatial derivative and $P(u)$ a polynomial in spatial derivatives of total order at most $α-1$, the solution cannot belong to the Gevrey class $G^z$ in time for any $z$ satisfying $1 \leq z<ασ$. |
| title | On the Gevrey regularity of the fifth-order Kadomtsev-Petviashvili-II equation: An improved approach |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.26669 |