On a class of systems of hyperbolic equations describing pseudo-spherical or spherical surfaces

TL;DR

This paper studies systems of hyperbolic PDEs that describe surfaces of constant curvature, providing classification results and new examples related to integrability conditions and known geometric systems.

Contribution

It offers a classification and characterization of hyperbolic PDE systems describing pseudospherical and spherical surfaces, including new generalized systems.

Findings

Classification of systems based on their geometric properties

New examples of integrable PDE systems with constant curvature surfaces

Connections to known models like Pohlmeyer-Lund-Regge and Konno-Oono systems

Abstract

We consider systems of partial differential equations of the form \begin{equation}\nonumber \left\{ \begin{array}{l} u_{xt}=F\left(u,u_x,v,v_x\right),\\ v_{xt}=G\left(u,u_x,v,v_x\right), \end{array} \right. \end{equation} describing pseudospherical (pss) or spherical surfaces (ss), meaning that, their generic solutions $u(x,t)\, v(x,t)$ provide metrics, with coordinates $(x,t)$, on open subsets of the plane, with constant curvature $K=-1$ or $K=1$. These systems can be described as the integrability conditions of $\mathfrak{g}$-valued linear problems, with $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{R})$ or $\mathfrak{g}=\mathfrak{su}(2)$, when $K=-1$, $K=1$, respectively. We obtain characterization and also classification results. Applications of the theory provide new examples and new families of systems of differential equations, which contain generalizations of a Pohlmeyer-Lund-Regge type system and of the Konno-Oono coupled dispersionless system.