Local isometric immersions of pseudospherical surfaces described by a class of third order partial differential equations

Abstract

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 $u_t-u_{xxt}=\lambda u^2{u}_{xxx}+G(u,u_x,u_{xx}),(\lambda \in \mathbb{R})$. 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.