Classification of Camassa-Holm-type differential systems describing pseudospherical or spherical surfaces

Mingyue Guo; Zhenhua Shi; Mingyue Guo; Zhenhua Shi

doi:10.3934/era.2026211

Electronic Research Archive

2026, Volume 34, Issue 7: 4777-4802. doi: 10.3934/era.2026211

Previous Article Next Article

Research article

Classification of Camassa-Holm-type differential systems describing pseudospherical or spherical surfaces

Mingyue Guo ¹,
Zhenhua Shi ^{1,2
,
,}

1.
School of Mathematics, Northwest University, Xi'an 710127, China
2.
Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

Received: 20 March 2026 Revised: 13 May 2026 Accepted: 28 May 2026 Published: 03 June 2026

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
$ \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 $ m, n\geq2 $ 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.
- Camassa-Holm-type equation,
- pseudospherical surfaces,
- nonlocal symmetry,
- finite symmetry transformation,
- bi-Hamiltonian structure
Citation: Mingyue Guo, Zhenhua Shi. Classification of Camassa-Holm-type differential systems describing pseudospherical or spherical surfaces[J]. Electronic Research Archive, 2026, 34(7): 4777-4802. doi: 10.3934/era.2026211

Related Papers:

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

$ \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 $ m, n\geq2 $ 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.

References

[1]	S. S. Chern, K. Tenenblat, Pseudospherical surfaces and evolution equations, Stud. Appl. Math., 74 (1986), 55–83. https://doi.org/10.1002/sapm198674155 doi: 10.1002/sapm198674155
[2]	R. Sasaki, Soliton equations and pseudospherical surfaces, Nucl. Phys. B, 154 (1979), 343–357. https://doi.org/10.1016/0550-3213(79)90517-0 doi: 10.1016/0550-3213(79)90517-0
[3]	E. Bour, Théorie de la déformation des surfaces, J. Éc. Polytech. Math., 19 (1862), 1–48.
[4]	D. C. Ferraioli, T. C. Silva, K. Tenenblat, A class of quasilinear second order partial differential equations which describe spherical or pseudospherical surfaces, J. Differ. Equations, 268 (2020), 7164–7182. https://doi.org/10.1016/j.jde.2019.11.069 doi: 10.1016/j.jde.2019.11.069
[5]	L. P. Jorge, K. Tenenblat, Linear problems associated with evolution equations of the type $u_tt = F(u, u_x, u_xx, u_t)$, Stud. Appl. Math., 77 (1987), 103–117. https://doi.org/10.1002/sapm1987772103 doi: 10.1002/sapm1987772103
[6]	V. P. G. Neto, Fifth order evolution equations describing pseudospherical surfaces, J. Differ. Equations, 249 (2010), 2822–2865. https://doi.org/10.1016/j.jde.2010.05.016 doi: 10.1016/j.jde.2010.05.016
[7]	E. G. Reyes, Equations of pseudo-spherical type (After S. S. Chern and K. Tenenblat), Results Math., 60 (2011), 53–101. https://doi.org/10.1007/s00025-011-0167-0 doi: 10.1007/s00025-011-0167-0
[8]	J. A. Cavalcante, K. Tenenblat, Conservation laws for nonlinear evolution equations, J. Math. Phys., 29 (1988), 1044–1049. https://doi.org/10.1063/1.528020 doi: 10.1063/1.528020
[9]	M. J. Ablowitz, D. J. Kaup, A. C. Newell, H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), 249–315. https://doi.org/10.1002/sapm1974534249 doi: 10.1002/sapm1974534249
[10]	Q. Ding, K. Tenenblat, On differential systems describing surfaces of constant curvature, J. Differ. Equations, 184 (2002), 185–214. https://doi.org/10.1006/jdeq.2001.4141 doi: 10.1006/jdeq.2001.4141
[11]	F. Kelmer, K. Tenenblat, On a class of systems of hyperbolic equations describing pseudospherical or spherical surfaces, J. Differ. Equations, 339 (2022), 372–394. https://doi.org/10.1016/j.jde.2022.08.017 doi: 10.1016/j.jde.2022.08.017
[12]	F. Kelmer, K. Tenenblat, Systems of differential equations of higher order describing pseudo-spherical or spherical surfaces, J. Differ. Equations, 424 (2025), 833–858. https://doi.org/10.1016/j.jde.2025.02.003 doi: 10.1016/j.jde.2025.02.003
[13]	F. Kelmer, On third-order evolution systems describing pseudo-spherical or spherical surfaces, preprint, arXiv: 2412.02657. https://doi.org/10.48550/arXiv.2412.02657
[14]	R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661–1664. https://doi.org/10.1103/PhysRevLett.71.1661 doi: 10.1103/PhysRevLett.71.1661
[15]	Z. J. Qiao, New integrable hierarchy its parametric solutions, cuspons, one-peak solutions, and M/W-shape peak solitons, J. Math. Phys., 48 (2007), 082701. https://doi.org/10.1063/1.2759830 doi: 10.1063/1.2759830
[16]	V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002. https://doi.org/10.1088/1751-8113/42/34/342002 doi: 10.1088/1751-8113/42/34/342002
[17]	D. C. Ferraioli, T. C. Silva, A class of third order quasilinear partial differential equations describing spherical or pseudospherical surfaces, J. Differ. Equations, 379 (2024), 524–568. https://doi.org/10.1016/j.jde.2023.10.032 doi: 10.1016/j.jde.2023.10.032
[18]	M. Y. Guo, J. Kang, Z. H. Shi, Z. W. Wu, On a class of third order differential equations describing pseudospherical or spherical surfaces, preprint, arXiv: 2508.20515. https://doi.org/10.48550/arXiv.2508.20515
[19]	T. C. Silva, K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Differ. Equations, 259 (2015), 4897–4923. https://doi.org/10.1016/j.jde.2015.06.016 doi: 10.1016/j.jde.2015.06.016
[20]	D. D. Holm, R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys. A: Math. Theor., 43 (2010), 492001. https://doi.org/10.1088/1751-8113/43/49/492001 doi: 10.1088/1751-8113/43/49/492001
[21]	A. N. W. Hone, V. Novikov, J. P. Wang, Two-component generalizations of the Camassa-Holm equation, Nonlinearity, 30 (2017), 622–658. https://doi.org/10.1088/1361-6544/aa5490 doi: 10.1088/1361-6544/aa5490
[22]	X. K. Chang, X. B. Hu, J. Szmigielski, Multipeakons of a two-component modified Camassa-Holm equation and the relation with the finite Kac-van Moerbeke lattice, Adv. Math., 299 (2016), 1–35. https://doi.org/10.1016/j.aim.2016.05.004 doi: 10.1016/j.aim.2016.05.004
[23]	J. F. Song, C. Z. Qu, Z. J. Qiao, A new integrable two-component system with cubic nonlinearity, J. Math. Phys., 52 (2011), 013503. https://doi.org/10.1063/1.3530865 doi: 10.1063/1.3530865
[24]	K. Tian, Q. P. Liu, Tri-Hamiltonian duality between the Wadati-Konno-Ichikawa hierarchy and the Song-Qu-Qiao hierarchy, J. Math. Phys., 54 (2013), 043513. https://doi.org/10.1063/1.4801858 doi: 10.1063/1.4801858
[25]	B. Q. Xia, Z. J. Qiao, A new two-component integrable system with peakon solutions, Proc. R. Soc. A, 471 (2015), 20140750. https://doi.org/10.1098/rspa.2014.0750 doi: 10.1098/rspa.2014.0750
[26]	K. Yan, Z. J. Qiao, Z. Y. Yin, Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions, Commun. Math. Phys., 336 (2015), 581–617. https://doi.org/10.1007/s00220-014-2236-1 doi: 10.1007/s00220-014-2236-1
[27]	B. Q. Xia, Z. J. Qiao, R. G. Zhou, A synthetical two-component model with peakon solutions, Stud. Appl. Math., 135 (2015), 248–276. https://doi.org/10.1111/sapm.12085 doi: 10.1111/sapm.12085
[28]	F. Galas, New non-local symmetries with pseudopotentials, J. Phys. A, 25 (1992), L981–L986. https://doi.org/10.1088/0305-4470/25/15/014 doi: 10.1088/0305-4470/25/15/014
[29]	G. A. Guthrie, Recursion operators and non-local symmetries, Proc. R. Soc. A, 446 (1994), 107–114. https://doi.org/10.1098/rspa.1994.0094 doi: 10.1098/rspa.1994.0094
[30]	H. Y. Zhou, K. Tian, N. H. Li, Four super integrable equations: nonlocal symmetries and applications, J. Phys. A: Math. Theor., 55 (2022), 225207. https://doi.org/10.1088/1751-8121/ac6a2b doi: 10.1088/1751-8121/ac6a2b
[31]	E. G. Reyes, Geometric integrability of the Camassa-Holm equation, Lett. Math. Phys., 59 (2002), 117–131. https://doi.org/10.1023/A:1014933316169 doi: 10.1023/A:1014933316169
[32]	N. H. Li, K. Tian, Nonlocal symmetries and Darboux transformations of the Camassa-Holm equation and modified Camassa-Holm equation revisited, J. Math. Phys., 63 (2022), 041501. https://doi.org/10.1063/5.0085540 doi: 10.1063/5.0085540
[33]	E. G. Reyes, On nonlocal symmetries of some shallow water equations, J. Phys. A, 40 (2007), 4467–4476. https://doi.org/10.1088/1751-8113/40/17/004 doi: 10.1088/1751-8113/40/17/004
[34]	Z. Q. Li, K. Tian, Nonlocal symmetries of two 2-component equations of Camassa-Holm type, Theor. Math. Phys., 220 (2024), 1471–1485. https://doi.org/10.1134/S0040577924090046 doi: 10.1134/S0040577924090046
[35]	M. Wadati, K. Konno, Y. H. Ichikawa, A generalization of inverse scattering method, J. Phys. Soc. Jpn., 6 (1979), 1965–1966. https://doi.org/10.1143/JPSJ.46.1965 doi: 10.1143/JPSJ.46.1965
[36]	M. Crampin, F. A. E. Pirani, D. C. Robinson, The soliton connection, Lett. Math. Phys., 2 (1977), 15–19. https://doi.org/10.1007/BF00420665
[37]	A. S. Fokas, On a class of physically important integrable equations, Physica D, 87 (1995), 145–150. https://doi.org/10.1016/0167-2789(95)00133-O doi: 10.1016/0167-2789(95)00133-O
[38]	B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Physica D, 95 (1996), 229–243. https://doi.org/10.1016/0167-2789(96)00048-6 doi: 10.1016/0167-2789(96)00048-6
[39]	P. J. Olver, P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E, 53 (1996), 1900–1906. https://doi.org/10.1103/PhysRevE.53.1900 doi: 10.1103/PhysRevE.53.1900
[40]	Z. J. Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys., 47 (2006), 112701. https://doi.org/10.1063/1.2365758 doi: 10.1063/1.2365758
[41]	A. S. Fokas, R. L. Anderson, On the use of isospectral eigenvalue problems for obtaining hereditary symmetries for Hamiltonian systems, J. Math. Phys., 23 (1982), 1066–1073. https://doi.org/10.1063/1.525495 doi: 10.1063/1.525495
[42]	A. S. Fokas, A symmetry approach to exactly solvable evolution equations, J. Math. Phys., 21 (1980), 1318–1325. https://doi.org/10.1063/1.524581 doi: 10.1063/1.524581
[43]	J. Kang, X. C. Liu, C. Z. Qu, On an integrable multi-component Camassa-Holm system arising from Möbius geometry, Proc. R. Soc. A, 477 (2021), 20210164. https://doi.org/10.1098/rspa.2021.0164 doi: 10.1098/rspa.2021.0164

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)