Local isometric immersions of pseudospherical surfaces described by a class of third order partial differential equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909623156473856 |
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| author | Guo, Mingyue Shi, Zhenhua |
| author_facet | Guo, Mingyue Shi, Zhenhua |
| contents | In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third order nonlinear partial differential equations with the type $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx}),(λ\in\mathbb{R})$. We prove that there is only two subclasses of equations admitting a local isometric immersion into the three dimensional Euclidean space $\mathbb{E}^3$ for which the coefficients of the second fundamental form depend on a jet of finite order of $u$, and furthermore, these coefficients are universal, namely, they are functions of $x$ and $t$, independent of $u$. Finally, we show that the generalized Camassa-Holm equation describing pseudospherical surfaces has a universal second fundamental form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19728 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local isometric immersions of pseudospherical surfaces described by a class of third order partial differential equations Guo, Mingyue Shi, Zhenhua Mathematical Physics In this paper, we study the problem of local isometric immersion of pseudospherical surfaces determined by the solutions of a class of third order nonlinear partial differential equations with the type $u_t - u_{xxt} = λu^2 u_{xxx} + G(u, u_x, u_{xx}),(λ\in\mathbb{R})$. We prove that there is only two subclasses of equations admitting a local isometric immersion into the three dimensional Euclidean space $\mathbb{E}^3$ for which the coefficients of the second fundamental form depend on a jet of finite order of $u$, and furthermore, these coefficients are universal, namely, they are functions of $x$ and $t$, independent of $u$. Finally, we show that the generalized Camassa-Holm equation describing pseudospherical surfaces has a universal second fundamental form. |
| title | Local isometric immersions of pseudospherical surfaces described by a class of third order partial differential equations |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2505.19728 |