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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.

MSC:
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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[1] M.-H. Amsler, ”Des surfaces a courbure negative constante dans l’espace a trois dimensions et de leurs singularites,” Math. Ann., 3, 234–256 (1955). · Zbl 0068.35102 · doi:10.1007/BF01343351
[2] V. A. Andreev and Yu. V. Brezhnev, ”Darboux transformation, positons and general superposition formula for the sine-Gordon equation,” Phys. Lett. A, 207, 58–66 (1995). · Zbl 1020.35517 · doi:10.1016/0375-9601(95)00663-N
[3] I. Ya. Bakel’man, A. L. Verner, and B. E. Kantor, Introduction to Differential Geometry ”in the Whole” [in Russian], Nauka, Moscow (1973).
[4] E. Beltrami, ”Sulla superficie di rotazione che serve di tipo alle superficie pseudosferiche,” Giorn. Mat., 10, Op. 2 (1872). · JFM 04.0406.03
[5] L. Bianchi, Lezioni di Geometria Differenziale, Bologna (1927).
[6] A. I. Bobenko, ”Surfaces in terms of (2 \(\times\) 2)-matrices. Old and new integrable cases” in: Harmonic Maps and Integrable Systems, Asp. Math., 23, Vieweg, Brunswick (1994). · Zbl 0841.53003
[7] A. I. Bobenko and A. V. Kitaev, ”On asymptotic cones of surfaces with constant curvature and the third Painlevé equation,” Manuscr. Math., 97, 489–516 (1998). · Zbl 0965.53010 · doi:10.1007/s002290050117
[8] F. J. Bureau, ”Differential equations with fixed critical points,” Ann. Math. Pur. Appl., 64, 229–364 (1964). · Zbl 0134.30601 · doi:10.1007/BF02410054
[9] I. V. Cherednik, ”On the realness conditions in finite-zone integration,” Dokl. Akad. Nauk SSSR,252, No. 5, 1104–1108 (1980).
[10] P. L. Chebyshev, ”Über den Schnitt von Bekleidungen,” Usp. Mat. Nauk, 1, No. 2, 38–42 (1946). · Zbl 0063.00746
[11] J. Cieslinski, ”The spectral interpretation of in-spaces of constant negative curvature immersed in \(\mathbb{R}\)2n,” Phys. Lett. A, 236, 425–430 (1997). · Zbl 0969.53503 · doi:10.1016/S0375-9601(97)00844-X
[12] J. Cieslinski, P. Goldstein, and A. Sym, ”Isothermic surfaces in \(\mathbb{E}^3 \) as soliton surfaces,” Phys. Lett. A, 205, No. 37–43 (1995). · Zbl 1020.53500
[13] G. Darboux, Lecons sur la Theorie Generale des Surfaces, Paris (1894).
[14] H. Dobriner, ”Die Flachen constanter Krummung mit einem System spharischer Krummungslinien dargestellt mit Hilfe von Thetafunctionen zweier Variabeln,” Acta Math., 9, 73–104 (1886–1887). · JFM 18.0425.01 · doi:10.1007/BF02406731
[15] B. A. Dubrovin, ”Theta-functions and nonlinear equations,” Usp. Mat. Nauk, 36, No. 2, 11–80 (1981). · Zbl 0478.58038
[16] B. A. Dubrovin, Riemann Surfaces and Nonlinear Equations [in Russian], Izhevsk (2001).
[17] B. A. Dubrovin and S. M. Natanzon, ”Real two-zone solutions of the sine-Gordon equation,” Funkts. Anal. Prilozh., 16, No. 1, 27–43 (1982). · Zbl 0554.35100
[18] N. V. Effimov, ”Surfaces with a slowly varying negative curvature,” Usp. Mat. Nauk, 21, No. 5, 3–58 (1966).
[19] A. Enneper, Analytisch-geometrische Untersuchungen, Gottinger Nachrichten (1868). · JFM 01.0227.03
[20] J. Fay, Theta-Functions on Riemann Surfaces, Lect. Notes Math., 352, Springer-Verlag (1973). · Zbl 0281.30013
[21] S. P. Finikov, Surface Theory [in Russian], Moscow-Leningrad (1934).
[22] A. Galini and T. A. Ivey, ”Backlund transformations and knots of constant torsion,” Knot Theory Ramifications, 7, No. 6, 719–746 (1998). · Zbl 0912.53002 · doi:10.1142/S0218216598000383
[23] I. V. Gribkov, ”Some solutions of the sine-Gordon equation obtained by the Backlund transformation,” Usp. Mat. Nauk, 33, No. 2, 191–192 (1978). · Zbl 0401.35095
[24] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, (1994). · Zbl 0836.14001
[25] D. Hilbert, Grundlagen der Geometrie, B. G. Teubner, Stuttgart (1999).
[26] T. Hoffmann, ”Discrete Amsler surfaces and a discrete Painlevé III equation,” in: Discrete Integrable Geometry and Physics, Oxford Lect. Ser. Math. Appl., 16, Clarendon Press. Oxford (1999), pp. 83–96. · Zbl 0944.53006
[27] E. L. Ince, Integration of Ordinary Differential Equations, Oliver & Boyd, Edinburgh (1939). · Zbl 0022.13601
[28] A. R. Its and V. Yu. Novokshenov, The Isomonodromic Deformation Method in the Theory of Painlevé Equations, Lect. Notes Math., 1191, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1986). · Zbl 0592.34001
[29] J. J. Klein, ”Geometric interpretation of the solutions of the sine-Gordon equation,” J. Math. Phys., 26, No. 9, 2181–2185 (1985). · doi:10.1063/1.526842
[30] V. A. Kozel and V. P. Kotlyarov, ”Finite-zone solutions of the sine-Gordon equation,” Preprint No. 9-77, FTINT, Khar’kov (1977).
[31] G. L. Lamb, Elements of Soliton Theory, John Wiley & Sons, New York, (1980). · Zbl 0445.35001
[32] E. V. Maevskii, ”Asymptotics of solutions to a second-order quasi-linear equation,” Comput. Math. Math. Phys. 38, No. 10, 1583–1590 (1998). · Zbl 0973.34042
[33] E. V. Maevskii, ”Two-soliton solutions of the sine-Gordon equation and the pseudospherical surfaces related to it,” Vestn. MGU, Ser. Fiz., Astron., 3 (2002). · Zbl 1260.35013
[34] E. V. Maevskii, ”On the Amsler pseudospherical surface,” Deposited at VINITI, No. 1695-V2002 (2002).
[35] E. V. Maevskii, ”Asymptotics of solutions of certain quasilinear second-order equations,” deposited at VINITI, No. 1696-V2002 (2002).
[36] V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons, Springer-Verlag, Berlin (1991). · Zbl 0744.35045
[37] D. Mumford, Tata Lectures on Theta. I: Introduction and Motivation: Theta Functions in One Variable. Basic Results on Theta Functions in Several Variables, Progr. Math., 28, Birkhauser, Boston-Basel-Stuttgart (1983). · Zbl 0509.14049
[38] S. Z. Nemeth, ”Backlund transformations of n-dimensional constant torsion curves,” Publ. Math. Debrecen, 53, Nos. 3–4, 271–279 (1998).
[39] A. P. Norden, Surface Theory [in Russian], Gostekhizdat, Moscow (1956).
[40] E. N. Pelinovskii, ”Certain exact methods in nonlinear wave theory,” Radiofizika, 19. Nos. 5–6 (1976).
[41] A. V. Pogorelov, Differential Geometry [in Russian], Nauka, Moscow (1974).
[42] A. G. Popov, ”A complete geometric interpretation of a one-soliton solution of an arbitrary amplitude of the sine-Gordon Equation,” Vestn. MGU, Mat., Mekh., 5, 3–8 (1990). · Zbl 0724.35094
[43] E. G. Poznyak, ”Geometric interpretation of regular solutions of the equation z xy = sin z,” Differents. Uravn., 15, No. 7, 1332–1336 (1979). · Zbl 0413.35063
[44] E. G. Poznyak, ”On a regular realization as a whole of two-dimensional metrics of negative curvature,” Ukr. Geom. Sb., 3, 78–92 (1966).
[45] E. G. Poznyak, ”Geometric studies related to the sine-Gordon Equation,” in: Progress in Science and Technology, Series on Problems in Geometry [in Russian], Vol. 8, All-Union Institute for Scientific and Technical Information, Moscow (1977), pp. 225–241.
[46] E. G. Poznyak and A. G. Popov, ”Geometry of the sine-Gordon equation,” in: Progress in Science and Technology, Series on Problems in Geometry [in Russian], Vol. 23, All-Union Institute for Scientific and Technical Information, Moscow (1991), pp. 99–130. · Zbl 0741.35072
[47] E. G. Poznyak and E. V. Shikin, Differential Geometry [in Russian], Moscow (1990).
[48] E. R. Rozendorn, ”Surfaces of negative curvature,” in: Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions [in Russian], Vol. 48, All-Union Institute for Scientific and Technical Information, Moscow (1989), pp. 98–195. · Zbl 0711.53004
[49] B. L. Rozhdestvenskii, ”A system of quasi-linear equations in the theory of surfaces,” Dokl. Akad. Nauk SSSR, 143, 50–52 (1962).
[50] E. V. Shilin, ”On isometric immersion of two-dimensional manifolds of negative curvature in the three-dimensional Euclidean space,” Mat. Zametki, 31, No. 4, 601–612 (1982). · Zbl 0491.53002
[51] G. E. Shilov, Mathematical Analysis. Functions of Several Variables [in Russian], Nauka, Moscow (1979). · Zbl 0497.26002
[52] V. I. Shulikovskii, Classical Differential Geometry in the Tensor Presentation [in Russian], Moscow (1963).
[53] A. O. Smirnov, ”Real elliptic solutions of the sine-Gordon equation,” Mat. Sb., 181, No. 6, 804–812 (1990). · Zbl 0705.35128
[54] A. O. Smirnov. ”3-Elliptic solutions of the sine-Gordon equation, Mat. Zametki, 62, No. 3, 440–452 (1997). · Zbl 0916.35106
[55] M. Spivak, Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus, Math. Monogr. Ser., W. A. Benjamin, New York-Amsterdam (1965). · Zbl 0141.05403
[56] R. Steuerwald, ”Über Ennepersche Flächen und Bäcklundsche Transformation,” Abh. Bayer. Akad. Wiss., N. F., 40, 1–106 (1936). · Zbl 0016.22402
[57] A. I. Sym, ”Soliton surfaces and their applications,” in: Geometric Aspects of the Einstein Equations and Integrable Systems, Lect. Notes Phys., 239, Springer-Verlag, Berlin (1985), pp. 154–231.
[58] F. Tricomi, Lezioni sulle Equazioni a Derivate Parziali, Editrice Gheroni, Torino (1954). · Zbl 0057.07502
[59] V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1976). · Zbl 0337.34014
[60] C. Wissler, ”Globale Tschbyscheff-Netze auf Riemannischen Mannigfaltigkeiten und Fortsetzung von Flächen konstanter negativer Krümmung,” Comment. Math. Helv., 47, No. 3, 348–372 (1972). · Zbl 0257.53003 · doi:10.1007/BF02566810
[61] S. A. Zadadaev, ”Travelling-wave-type solutions of the sine-Gordon equation and pseudospherical surfaces,” Vestn. MGU, Mat., Mekh., 2 (1994). · Zbl 0879.35141
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