标题： 关于描述伪球面或球面的三阶演化系统的研究 显示英文标题

标题： On Third-Order Evolution Systems Describing Pseudo-Spherical or Spherical Surfaces

摘要： 我们考虑一类形式为\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*}的三阶演化方程，描述伪球面（\textbf{pss}）或球面（\textbf{ss}），这意味着，它们的通解$(u(x,t), v(x,t))$在平面的开子集上提供坐标为$(x,t)$的度量，具有常曲率$K=-1$或$K=1$。 这些系统可以被描述为当参数分别为$K=-1$和$K=1$时，$\mathfrak{g}$-值线性问题的可积性条件，分别对应$\mathfrak{g}=\mathfrak{sl}(2,\R)$或$\mathfrak{g}=\mathfrak{su}(2)$。我们得到了系统的特征化及分类结果。这些结果的应用提供了此类系统的新例子和新族类，其中包括耦合 KdV 方程和 mKdV 型方程以及非线性 Schrödinger 方程。此外，该理论还用于推导耦合 KdV 系统的 Bäcklund 变换。 显示英文摘要

摘要： 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 Schr\"odinger equations. Additionally, this theory is applied to derive a B\"acklund transformation for the coupled KdV system.