@article{Guo2026, 
author = {Mingyue Guo and Zhenhua Shi},
title = {Classification of Camassa-Holm-type differential systems describing pseudospherical or spherical surfaces},
year = {2026},
journal = {Electronic Research Archive},
volume = {34},
number = {7},
pages = {4777-4802},
keywords = {Camassa-Holm-type equation, pseudospherical surfaces, nonlocal symmetry, finite symmetry transformation, bi-Hamiltonian structure},
url = {https://www.sciopen.com/article/10.3934/era.2026211},
doi = {10.3934/era.2026211},
abstract = {In this paper, we study nonlinear partial differential systems that describe surfaces of constant curvature. From the flatness condition of connection 1-forms, we present a classification of Camassa-Holm-type systems of the form         {                                        u                          t                                −                      u                          x              x              t                                                          =          F          (          x          ,          t          ,          u          ,                      u                          x                                ,          …          ,                      ∂                          m                                u                      /                                ∂            x                          m                                ,          v          ,                      v                          x                                ,          …          ,                      ∂                          n                                v                      /                                ∂            x                          n                                )          ,                                                  v                          t                                −                      v                          x              x              t                                                          =          G          (          x          ,          t          ,          u          ,                      u                          x                                ,          …          ,                      ∂                          m                                u                      /                                ∂            x                          m                                ,          v          ,                      v                          x                                ,          …          ,                      ∂                          n                                v                      /                                ∂            x                          n                                )          ,                        with    m  ,  n  ≥  2 and    F,    G smooth functions, describing pseudospherical or spherical surfaces. We also establish classification results for a special type of third-order system. Applications of these results provide new examples of such systems, including the Song-Qu-Qiao system, the Xia-Qiao-Zhou system, and the two-component modified Camassa-Holm system. Furthermore, we construct the nonlocal symmetry for the Xia-Qiao-Zhou system from the gradients of the spectral parameter. By introducing an appropriate pseudo-potential, we prolong the nonlocal symmetry to an enlarged system and calculate the corresponding finite symmetry transformation. On this basis, we derive nontrivial solutions to the Xia-Qiao-Zhou system.}
}