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arXiv · 2508.20515

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

Mingyue Guo·Jing Kang·Zhenhua Shi·Zhiwei Wu
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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.

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Mingyue Guo, Jing Kang, Zhenhua Shi, Zhiwei Wu. 2025-08-28. On a class of third order differential equations describing pseudospherical or spherical surfaces. https://arxiv.org/abs/2508.20515

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