Hodgkin, Luke Soliton equations and hyperbolic maps. (English) Zbl 0538.35069 Ann. Inst. Henri Poincaré, Sect. A 38, 49-58 (1983). 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 Keywords:soliton equations; hyperbolic maps; nonlinear evolution equation; metric of curvature; scattering equation; sine-Gordon equations; Bäcklund transformation PDF BibTeX XML Cite \textit{L. Hodgkin}, Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. A 38, 49--58 (1983; Zbl 0538.35069) Full Text: Numdam EuDML References: [1] M. Crampin , F.A.E. Pirani and D.C. Robinson , Lett. Math. Phys. , t. 2 , 1977 , p. 15 . MR 489530 | Zbl 0363.35032 · Zbl 0363.35032 · doi:10.1007/BF00420665 [2] M. Crampin , L. Hodgkin , P.J. Mccarthy and D.C. Robinson , 2-manifolds of constant curvature, 3-parameter isometry groups and Bäcklund transformations , to appear in Rep. Math. Phys. Zbl 0496.35074 · Zbl 0496.35074 · doi:10.1016/0034-4877(80)90005-1 [3] R. Hermann , The geometry of non-linear differential equations, Bäcklund transformations and solitons , Part A ( Math. Sci. Press , Brookline, MA , 1976 ). MR 442961 | Zbl 0367.35001 · Zbl 0367.35001 [4] M. Crampin , Phys. Lett. t. 66A , 1978 , p. 170 . MR 598750 [5] R. Sasaki and R.K. Bullough , in Nonlinear Evolution Equations and Dynamical Systems (ed. Boiti et al.), Lecture Notes in Physics , no. 120 ( Springer , Berlin , Heidelberg , New York , 1980 ). MR 581888 [6] M.J. Ablowitz , D.J. Kaup , A.C. Newell and H. Segur , Phys. Rev. Lett. , t. 31 , 1973 , p. 125 . MR 406176 [7] L.P. Eisenhart , A. Treatise on the Differential Geometry of Curves and Surfaces ( Dover , New York , 1960 ). MR 115134 | Zbl 0090.37803 · Zbl 0090.37803 [8] F.A.E. Pirani , in Nonlinear Evolution Equations and Dynamical Systems (ed. Boiti et al.), Lecture Notes in Physics , no. 120 ( Springer , Berlin , Heidelberg , New York , 1980 ). MR 581897 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.