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

TL;DR

This paper classifies third order PDEs related to pseudospherical surfaces that admit local isometric immersions into Euclidean space, showing only two subclasses have universal second fundamental forms, including the generalized Camassa-Holm equation.

Contribution

It identifies specific subclasses of third order PDEs that allow for universal local isometric immersions of pseudospherical surfaces into three-dimensional space.

Findings

Only two subclasses of equations admit such immersions.

The second fundamental form coefficients are universal functions of space and time.

The generalized Camassa-Holm equation has a universal second fundamental form.

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.