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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.02657 |
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| _version_ | 1866917855841222656 |
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| author | Kelmer, Filipe |
| author_facet | Kelmer, Filipe |
| contents | We consider a class of third-order evolution equations of the form \begin{equation*} \left\{
\begin{array}{l}
\displaystyle u_{t}=F\left(x,t,u,u_x,u_{xx},u_{xxx},v,v_x,v_{xx},v_{xxx}\right),
\displaystyle v_{t}=G\left(x,t,u,u_x,u_{xx},u_{xxx},v,v_x,v_{xx},v_{xxx}\right),
\end{array}
\right. \end{equation*} describing pseudos-pherical (\textbf{pss}) or spherical surfaces (\textbf{ss}), meaning that, their generic solutions $(u(x,t), v(x,t))$ provide metrics, with coordinates $(x,t)$, on open subsets of the plane, with constant curvature $K=-1$ or $K=1$. These systems can be described as the integrability conditions of $\mathfrak{g}$-valued linear problems, with $\mathfrak{g}=\mathfrak{sl}(2,\R)$ or $\mathfrak{g}=\mathfrak{su}(2)$, when $K=-1$, $K=1$, respectively. We obtain characterization and also classification results. Applications of these results provide new examples and new families of such systems, which also contain systems of coupled KdV and mKdV-type equations and nonlinear Schrödinger equations. Additionally, this theory is applied to derive a Bäcklund transformation for the coupled KdV system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02657 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Third-Order Evolution Systems Describing Pseudo-Spherical or Spherical Surfaces Kelmer, Filipe Differential Geometry Analysis of PDEs We consider a class of third-order evolution equations of the form \begin{equation*} \left\{ \begin{array}{l} \displaystyle u_{t}=F\left(x,t,u,u_x,u_{xx},u_{xxx},v,v_x,v_{xx},v_{xxx}\right), \displaystyle v_{t}=G\left(x,t,u,u_x,u_{xx},u_{xxx},v,v_x,v_{xx},v_{xxx}\right), \end{array} \right. \end{equation*} describing pseudos-pherical (\textbf{pss}) or spherical surfaces (\textbf{ss}), meaning that, their generic solutions $(u(x,t), v(x,t))$ provide metrics, with coordinates $(x,t)$, on open subsets of the plane, with constant curvature $K=-1$ or $K=1$. These systems can be described as the integrability conditions of $\mathfrak{g}$-valued linear problems, with $\mathfrak{g}=\mathfrak{sl}(2,\R)$ or $\mathfrak{g}=\mathfrak{su}(2)$, when $K=-1$, $K=1$, respectively. We obtain characterization and also classification results. Applications of these results provide new examples and new families of such systems, which also contain systems of coupled KdV and mKdV-type equations and nonlinear Schrödinger equations. Additionally, this theory is applied to derive a Bäcklund transformation for the coupled KdV system. |
| title | On Third-Order Evolution Systems Describing Pseudo-Spherical or Spherical Surfaces |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2412.02657 |