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Dokl. Akad. Nauk SSSR, 1979, Volume 247, Number 5, Pages 1103–1107 (Mi dan42919)  

This article is cited in 45 scientific papers (total in 45 papers)


The Klein–Gordon equation with nontrivial group

A. V. Zhiber, A. B. Shabat

Ufa Aviation University named after Sergo Orgonikidze

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UDC: 517.43+519.46
Presented: В. С. Владимиров
Received: 05.02.1979

Citation: A. V. Zhiber, A. B. Shabat, “The Klein–Gordon equation with nontrivial group”, Dokl. Akad. Nauk SSSR, 247:5 (1979), 1103–1107

Citation in format AMSBIB
\by A.~V.~Zhiber, A.~B.~Shabat
\paper The Klein--Gordon equation with nontrivial group
\jour Dokl. Akad. Nauk SSSR
\yr 1979
\vol 247
\issue 5
\pages 1103--1107

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    This publication is cited in the following articles:
    1. V. V. Sokolov, A. B. Shabat, “$(L,A)$-Pairs and a Ricatti type substitution”, Funct. Anal. Appl., 14:2 (1980), 148–150  mathnet  crossref  mathscinet  zmath
    2. A. N. Leznov, M. V. Saveliev, “Two-dimensional nonlinear system of differential equations $x_{\alpha,z\overline{z}}=\exp(kx)_\alpha$”, Funct. Anal. Appl., 14:3 (1980), 238–240  mathnet  crossref  mathscinet  zmath  isi
    3. B. S. Getmanov, “Integrable two-dimensional Lorentz-invariant nonlinear model of a complex scalar field (complex sine-Gordon II)”, Theoret. and Math. Phys., 48:1 (1981), 572–579  mathnet  crossref  mathscinet  isi
    4. A. N. Leznov, V. G. Smirnov, A. B. Shabat, “The group of internal symmetries and the conditions of integrability of two-dimensional dynamical systems”, Theoret. and Math. Phys., 51:1 (1982), 322–330  mathnet  crossref  mathscinet  zmath  isi
    5. A. A. Belavin, V. G. Drinfeld, “Solutions of the classical Yang–Baxter equation for simple Lie algebras”, Funct. Anal. Appl., 16:3 (1982), 159–180  mathnet  crossref  mathscinet  zmath  isi
    6. A. G. Meshkov, “Symmetries of scalar fields. I”, Theoret. and Math. Phys., 55:2 (1983), 445–450  mathnet  crossref  mathscinet  zmath  isi
    7. M. B. Sheftel, “On the infinite-dimensional noncommutative Lie–Bäcklund algebra associated with the equations of one-dimensional gas dynamics”, Theoret. and Math. Phys., 56:3 (1983), 878–891  mathnet  crossref  mathscinet  zmath  isi
    8. A. G. Meshkov, “Symmetries of scalar fields. II”, Theoret. and Math. Phys., 57:3 (1983), 1209–1216  mathnet  crossref  mathscinet  zmath  isi
    9. V. V. Nesterenko, “Nonlinear $\sigma$ model for the Dodd–Bullough equation”, Theoret. and Math. Phys., 58:2 (1984), 126–131  mathnet  crossref  mathscinet  zmath  isi
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    11. V. Yu. Novokshenov, “Asymptotics as $t\to\infty$ of the solution of the Cauchy problem for a two-dimensional generalization of the Toda lattice”, Math. USSR-Izv., 24:2 (1985), 347–382  mathnet  crossref  mathscinet  zmath
    12. V. S. Vladimirov, I. V. Volovich, “Local and nonlocal currents for nonlinear equations”, Theoret. and Math. Phys., 62:1 (1985), 1–20  mathnet  crossref  mathscinet  zmath  isi
    13. A. V. Mikhailov, A. B. Shabat, “Integrability conditions for systems of two equations of the form $u_t+A(u)u_{xx}+F(u,u_x)$. I”, Theoret. and Math. Phys., 62:2 (1985), 107–122  mathnet  crossref  mathscinet  zmath  isi
    14. A. N. Leznov, “Simplest soliton solutions of the equations of the periodic Toda chain of the series $A_k$ in the formalism of the scalar $L$$A$ pair”, Theoret. and Math. Phys., 71:3 (1987), 598–605  mathnet  crossref  mathscinet  isi
    15. A. V. Mikhailov, A. B. Shabat, R. I. Yamilov, “The symmetry approach to the classification of non-linear equations. Complete lists of integrable systems”, Russian Math. Surveys, 42:4 (1987), 1–63  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. F. Kh. Mukminov, “On straightening the characteristics of a quasilinear second-order equation”, Theoret. and Math. Phys., 75:1 (1988), 340–345  mathnet  crossref  mathscinet  zmath  isi
    17. B. V. Lidskii, E. I. Shulman, “Periodic solutions of the equation $u_{tt}-u_{xx}+u^3=0$”, Funct. Anal. Appl., 22:4 (1988), 332–333  mathnet  crossref  mathscinet  zmath  isi
    18. V. A. Andreev, “Bäcklund transformations of the Bullough–Dodd–Zhiber–Shabat equation and symmetries of integrable equations”, Theoret. and Math. Phys., 79:1 (1989), 448–450  mathnet  crossref  mathscinet  isi
    19. I. Yu. Cherdantsev, R. A. Sharipov, “Finite-gap solutions of the Bullough–Dodd–Zhiber–Shabat equation”, Theoret. and Math. Phys., 82:1 (1990), 108–11  mathnet  crossref  mathscinet  zmath  isi
    20. Yu. A. Markov, “A class of exact solutions for a kinetic model of an equilibrium plasma”, Theoret. and Math. Phys., 91:1 (1992), 418–427  mathnet  crossref  mathscinet  isi
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    22. A. V. Zhiber, “Quasilinear hyperbolic equations with an infinite-dimensional symmetry algebra”, Russian Acad. Sci. Izv. Math., 45:1 (1995), 33–54  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    23. E. I. Ganzha, “Osnovnye serii neodnorodnykh sistem gidrodinamicheskogo tipa s postoyannymi matritsami, obladayuschikh zakonami sokhraneniya 1-go poryadka”, Fundament. i prikl. matem., 1:3 (1995), 641–648  mathnet  mathscinet  zmath
    24. E. V. Kopanev, S. V. Kryukov, M. A. Suhoruchkin, “Conservation currents in quantum integrable models”, Theoret. and Math. Phys., 113:1 (1997), 1235–1243  mathnet  crossref  crossref  mathscinet  zmath  isi
    25. V. E. Adler, S. Ya. Startsev, “Discrete analogues of the Liouville equation”, Theoret. and Math. Phys., 121:2 (1999), 1484–1495  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    26. V. E. Adler, A. B. Shabat, R. I. Yamilov, “Symmetry approach to the integrability problem”, Theoret. and Math. Phys., 125:3 (2000), 1603–1661  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    27. A. V. Zhiber, V. V. Sokolov, “Exactly integrable hyperbolic equations of Liouville type”, Russian Math. Surveys, 56:1 (2001), 61–101  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    28. A. B. Borisov, S. A. Zykov, M. V. Pavlov, “Tzitzéica Equation and Proliferation of Nonlinear Integrable Equations”, Theoret. and Math. Phys., 131:1 (2002), 550–557  mathnet  crossref  crossref  mathscinet  zmath  isi
    29. S. D. Griffiths, R. Grimshaw, K. R. Khusnutdinova, “The Influence of Modulational Instability on Energy Exchange in Coupled Sine-Gordon Equations”, Theoret. and Math. Phys., 137:1 (2003), 1448–1458  mathnet  crossref  crossref  mathscinet  zmath  isi
    30. A. V. Kiselev, “Methods of geometry of differential equations in analysis of integrable models of field theory”, J. Math. Sci., 136:6 (2006), 4295–4377  mathnet  crossref  mathscinet  zmath  elib  elib
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    32. A. V. Zhiber, R. D. Murtazina, “On the characteristic Lie algebras for equations $u_{xy}=f(u,u_x)$”, J. Math. Sci., 151:4 (2008), 3112–3122  mathnet  crossref  mathscinet  zmath  elib  elib
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    34. B. I. Suleimanov, ““Quantizations” of the second Painlevé equation and the problem of the equivalence of its $L$$A$ pairs”, Theoret. and Math. Phys., 156:3 (2008), 1280–1291  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    36. A. G. Meshkov, V. V. Sokolov, “Hyperbolic equations with third-order symmetries”, Theoret. and Math. Phys., 166:1 (2011), 43–57  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    37. B. I. Suleimanov, ““Kvantovaya” linearizatsiya uravnenii Penleve kak komponenta ikh $L,A$ par”, Ufimsk. matem. zhurn., 4:2 (2012), 127–135  mathnet
    38. A. V. Zhiber, R. D. Murtazina, I. T. Khabibullin, A. B. Shabat, “Kharakteristicheskie koltsa Li i integriruemye modeli matematicheskoi fiziki”, Ufimsk. matem. zhurn., 4:3 (2012), 17–85  mathnet  mathscinet
    39. M. N. Kuznetsova, “O nelineinykh giperbolicheskikh uravneniyakh, svyazannykh differentsialnymi podstanovkami s uravneniem Kleina–Gordona”, Ufimsk. matem. zhurn., 4:3 (2012), 86–103  mathnet  mathscinet
    40. A. G. Meshkov, V. V. Sokolov, “Integriruemye evolyutsionnye uravneniya s postoyannoi separantoi”, Ufimsk. matem. zhurn., 4:3 (2012), 104–154  mathnet
    41. A. U. Sakieva, “Kharakteristicheskoe koltso Li uravneniya Zhibera–Shabata–Tsitseiki”, Ufimsk. matem. zhurn., 4:3 (2012), 155–160  mathnet
    42. S. V. Sazonov, N. V. Ustinov, “Extremely short vector solitons under the conditions of conical refraction”, JETP Letters, 99:9 (2014), 503–507  mathnet  crossref  crossref  isi  elib  elib
    43. V. G. Marikhin, “Action as an invariant of Bäcklund transformations for Lagrangian systems”, Theoret. and Math. Phys., 184:1 (2015), 953–960  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    44. D. V. Millionshchikov, “Characteristic Lie algebras of the sinh-Gordon and Tzitzeica equations”, Russian Math. Surveys, 72:6 (2017), 1174–1176  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    45. S. Ya. Startsev, “Structure of set of symmetries for hyperbolic systems of Liouville type and generalized Laplace invariants”, Ufa Math. J., 10:4 (2018), 103–110  mathnet  crossref  isi
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