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

TL;DR

This work addresses the problem of local isometric immersion for pseudospherical surfaces described by the third-order PDE $u_t - u_{xxt} = \lambda u^2 u_{xxx} + G(u,u_x,u_{xx})$. It shows that, among such equations, only two families admit a local immersion into $\mathbb{E}^3$ with the second fundamental form coefficients depending on a finite jet of $u$, and crucially these coefficients are universal, i.e., functions of $x$ and $t$ only. The generalized Camassa-Holm equation for pseudospherical surfaces is shown to possess a universal second fundamental form, highlighting a special geometric status among the class. The results connect Cartan's structure equations and the Gauss-Codazzi framework to a sharp PDE classification, clarifying when the extrinsic geometry can be controlled independently of the solution.

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

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• Lemma 3.2
• Proof