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Intrinsic formulation of geometric integrability and associated Riccati system generating conservation laws. (English) Zbl 1188.53090

The aim of the paper is to study, firstly, the formulation of Bäcklund transformations based on a Pfaffian system for the case of nonlinear evolution equations which describe pseudospherical surfaces, this is, surfaces with negative constant Gauss curvature, and secondly the determination of conservation laws for such equations.

Starting from the structure equations of a surface with Gauss curvature equal to -1, the author is able to transform them into an associated system of differential equations in a Riccati form and to formulate the equivalent linear problem. All this has been done in an intrinsic way.

Finally, it is shown that geometrical properties of a pseudospherical surface provide a systematic method for obtaining an infinite number of conservation laws.

MSC:
53C80Applications of global differential geometry to physics
53C21Methods of Riemannian geometry, including PDE methods; curvature restrictions (global)
35Q53KdV-like (Korteweg-de Vries) equations
53A10Minimal surfaces, surfaces with prescribed mean curvature