On a class of third order differential equations describing pseudospherical or spherical surfaces

In this paper, we study third order nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness of connection 1-forms, we present a classification of equations with the type $u_t-u_{xxt}=\lambda u^2{u}_{xxx}+G(u,u_x,u_{xx})(\lambda \in \mathbb{R})$, which describe pseudospherical or spherical surfaces. We show that series of typical soliton equations belong to certain subclass, such as the generalized Camassa-Holm equation, which gives a geometric explanation to these equations.