# On a class of third order differential equations describing pseudospherical or spherical surfaces

## TL;DR

This paper classifies third order nonlinear PDEs that describe surfaces of constant curvature, linking them to soliton equations like the generalized Camassa-Holm, and providing geometric insights into their structure.

## Contribution

It introduces a classification of third order PDEs based on flat connection forms, connecting them to known soliton equations and geometric surface descriptions.

## Key findings

- Classification of equations describing pseudospherical and spherical surfaces
- Identification of soliton equations within the classified family
- Geometric interpretation of generalized Camassa-Holm equation

## Abstract

In this paper, we study third order nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness of connection 1-forms, we present a classification of equations with the type $u_t - u_{xxt} = \lambda u^2 u_{xxx} + G(u, u_x, u_{xx}) (\lambda\in\mathbb{R})$, which describe pseudospherical or spherical surfaces. We show that series of typical soliton equations belong to certain subclass, such as the generalized Camassa-Holm equation, which gives a geometric explanation to these equations.