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

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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

$\left\{\begin{array}{l} u_{t}-u_{x x t}=F\left(x, t, u, u_{x}, \ldots, \partial^{m} u / \partial_{x}^{m}, v, v_{x}, \ldots, \partial^{n} v / \partial_{x}^{n}\right), \\ v_{t}-v_{x x t}=G\left(x, t, u, u_{x}, \ldots, \partial^{m} u / \partial_{x}^{m}, v, v_{x}, \ldots, \partial^{n} v / \partial_{x}^{n}\right), \end{array}\right.$

withm,n ≥ 2 andF,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.

Keywords

Camassa-Holm-type equation / pseudospherical surfaces / nonlocal symmetry / finite symmetry transformation / bi-Hamiltonian structure

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