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Conservation laws and Calapso-Guichard deformations of equations describing pseudo-spherical surfaces. (English) Zbl 0992.53005
From author’s abstract: The relation between the Chern and Tenenblat approach to conservation laws of equations describing pseudo-spherical surfaces (conservation laws obtained from pseudo-spherical structure) and the more familiar “Riccati equation” approach (conservation laws obtained from associated linear problem) is investigated. Two examples (cylindrical Korteweg-de Vries (KdV) and Lund-Regge equations) are presented. Chern and Tenenblat’s point of view is then connected with the theory of soliton surfaces. A generalization of the original Chern-Tenenblat construction of conservation law-results and a reasonable family of large deformations for scalar equations describing pseudo-spherical surfaces, the “equations describing Calapso-Guichard surfaces”, can be introduced. It is shown that these equations are also the integrability conditions of linear problems.
MSC:
53A05Surfaces in Euclidean space
37K25Relations of infinite-dimensional systems with differential geometry
35L65Conservation laws
37K40Soliton theory, asymptotic behavior of solutions
35Q53KdV-like (Korteweg-de Vries) equations