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Soliton equations and hyperbolic maps. (English) Zbl 0538.35069
Summary: A solution of the AKNS scattering equation associated to a nonlinear evolution equation determines an isometry from \(({\mathbb{R}}^ e,g)\) to the hyperbolic plane H, where g is the metric of curvature -1 defined by the scattering equation. This correspondence is (locally) 2-1 from solutions to isometries. For the modified KdV and sine-Gordon equations, the scattering equations can be seen as a flow on the space of constant-speed curves in H, with a simply-described curvature function. A geometrical interpretation of the Bäcklund transformation is given, together with a ”soliton” example.
MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35P25 Scattering theory for PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
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References:
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