Systems of partial differential equations describing pseudo-spherical or spherical surfaces
Abstract
In this paper, we study systems of nonlinear partial differential equations which describe surfaces of constant curvature. From the flatness condition of connection 1-forms, we present a classification of systems of Camassa-Holm-type equations of the form \begin{equation*} \left\{ \begin{aligned} u_{t} - u_{xxt} &= F(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \\ v_{t} - v_{xxt} &= G(x, t, u, u_{x}, \dots, \partial ^{m} u/\partial _x^{m}, v, v_{x}, \dots, \partial ^{n} v/\partial _x^{n}), \end{aligned} \right. \end{equation*} with , for and smooth functions, describing pseudospherical or spherical surfaces. We also establish classification results for a special type of third-order system. Applications of the results provide new examples of such systems, such as the Song-Qu-Qiao system, the two-component Camassa-Holm system with cubic nonlinearity, and the modified Camass-Holm-type system. Moreover, we construct the nonlocal symmetry and a non-trivial solutions for the two-component Camassa-Holm system with cubic nonlinearity from the gradients of spectral parameters.
Cite
@article{arxiv.2603.11674,
title = {Systems of partial differential equations describing pseudo-spherical or spherical surfaces},
author = {Mingyue Guo and Jing Kang and Zhenhua Shi},
journal= {arXiv preprint arXiv:2603.11674},
year = {2026}
}