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Analytical approaches to the study of the sine-Gordon equation and pseudospherical surfaces. (English. Russian original) Zbl 1130.53012
J. Math. Sci., New York 142, No. 5, 2377-2418 (2007); translation from Sovrem. Mat. Prilozh. 31, 13-52 (2005).
This is a very good and detailed survey on the analytical methods of the study of the sine-Gordon equation and their applications to the theory of pseudospherical surfaces. Concerning the methods of integrating this equation it covers not only classical results obtained in the 19th century but also some very new approaches, in particular, based on the modern theory of integrable systems. The same is valid for the part concerning applications to surface theory. The expositions of results are, in general, supplied by detailed derivations and this article may be recommended as a valuable source of information on this subject.

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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