Third order differential equations and local isometric immersions of pseudospherical surfaces

TL;DR

This paper investigates third-order differential equations describing pseudospherical surfaces, focusing on conditions under which their local isometric immersions in three-dimensional space have coefficients depending only on finite jets of solutions, extending classical results known for the sine-Gordon equation.

Contribution

It demonstrates that for a class of third-order equations including Camassa-Holm and Degasperis-Procesi, the second fundamental form coefficients are universal if they depend on finite jets of solutions.

Findings

Coefficients of the second fundamental form are independent of specific solutions.

The results extend classical properties of the sine-Gordon equation to other integrable equations.

The paper characterizes when local isometric immersions have solution-independent coefficients.

Abstract

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the presence associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa-Holm and Degasperis-Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in ${\bf E}^{3}$ of the pseudospherical surfaces defined by the solutions of equations belonging to the class of Chern and Tenenblat. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the pde. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In the present paper, we consider third-order equations of the form $u_{t}-u_{xxt}=\lambda u u_{xxx} + G(u,u_x,u_{xx}),\, \lambda \in \mathbb{R}, $ which describe pseudospherical surfaces. This class contains the Camassa-Holm and Degasperis-Procesi equations as special cases. We show that whenever there exists a local isometric immersion in ${\bf E}^3$ for which the coefficients of the second fundamental form depend on a jet of finite order of $u$, then these coefficients are universal in the sense of being independent on the choice of solution $u$.