Abstract Second order partial differential equations which describe spherical surfaces (ss) or pseudospherical surfaces (pss) are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature K = 1 or K = − 1 , respectively, and they can be seen as the compatibility condition of an associated su ( 2 ) -valued or sl ( 2 , R ) -valued linear problem, also referred to as a zero curvature representation. Under certain assumptions we give a complete and explicit classification of equations of the form z t t = A ( z , z x , z t ) z x x + B ( z , z x , z t ) z x t + C ( z , z x , z t ) describing pss or ss, in terms of some arbitrary differentiable functions. Several examples of such equations are provided by choosing the arbitrary functions. In particular, well known equations which describe pseudospherical surfaces, such as the short-pulse and the constant astigmatism equations, as well as their generalizations and their spherical analogues are included in the paper.

中文翻译：

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一类描述球面或拟球面的拟线性二阶偏微分方程

摘要 考虑了描述球面（ss）或拟球面（pss）的二阶偏微分方程。这些方程分别等价于高斯曲率 K = 1 或 K = − 1 的度量的结构方程，它们可以看作是关联的 su ( 2 ) 值或 sl ( 2 , R ) 的兼容性条件值线性问题，也称为零曲率表示。在某些假设下，我们给出了形式为 ztt = A ( z , zx , zt ) zxx + B ( z , zx , zt ) zxt + C ( z , zx , zt ) 描述 pss 或 ss 的方程的完整而明确的分类，在一些任意可微函数方面。通过选择任意函数提供了此类方程的几个示例。特别是，