Equations of pseudo-spherical type (After S. S. Chern and K. Tenenblat)
Hierarchies of evolution equations of pseudo-spherical type are introduced, generalizing the notion of a single equation describing pseudo-spherical surfaces due to S. S. Chern and Keti Tenenblat. This new structure provides a link between differential geometry and the study of hierarchies of equations which are the integrability condition of \(sl(2;\mathbb R)\)-valued linear problems. The purpose of this paper is to survey and analyze some recent developments around the notion of a scalar partial differential equation describing pseudo-spherical surfaces, including recent results. It is shown how conservation laws, pseudo-potentials and linear problems arise naturally from geometric considerations, and it is also explained how Darboux and Bäcklund transformations can be constructed starting from geometric data. Classification results for equations in this class are stated, and hierarchies of equations of pseudo-spherical type are introduced, providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of \(sl(2,\mathbb R)\)-valued linear problems. Furthermore, the existence of correspondences between any two solutions to equations of pseudo-spherical type is reviewed, and a correspondence theorem for hierarchies is also mentioned. As applications, an elementary immersion result for pseudo-spherical metrics arising from the Chern-Tenenblat construction is proven, and non-local symmetries of the Kaup-Kupershmidt, Sawada-Kotera, fifth-order Korteweg-de Vries and Camassa-Holm (CH) equations with non-zero critical wave speed are considered. It is shown that the existence of a non-local symmetry of a particular type is enough to single the first three equations out from a whole family of equations describing pseudo-spherical surfaces, while, in the CH case, it is shown that it admits an infinite-dimensional Lie algebra of non-local symmetries which includes the Virasoro algebra. The final part of the paper is devoted to nonlocal symmetries of the Camassa-Holm equation.
scientific article; zbMATH DE number 6104266
scientific article