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SIGMA 8 (2012), 090, 37 pages arXiv:1111.7255
https://doi.org/10.3842/SIGMA.2012.090
Contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”
The Klein-Gordon Equation and Differential Substitutions of the Form $\boldsymbol{v=\varphi(u,u_x,u_y)}$
Mariya N. Kuznetsova a, Aslı Pekcan b and Anatoliy V. Zhiber c
a) Ufa State Aviation Technical University, 12 K. Marx Str., Ufa, Russia
b) Department of Mathematics, Istanbul University, Istanbul, Turkey
c) Ufa Institute of Mathematics, Russian Academy of Science, 112 Chernyshevskii Str., Ufa, Russia
Received April 25, 2012, in final form November 14, 2012; Published online November 26, 2012
Abstract
We present the complete classification of equations of the form $u_{xy} = f(u, u_x, u_y)$
and the Klein-Gordon equations $v_{xy} = F(v)$ connected with one another
by differential substitutions $v = \varphi(u, u_x, u_y)$ such that
$\varphi_{u_x}\varphi_{u_y}\neq 0$ over the ring of complex-valued variables.
Key words:
Klein-Gordon equation; differential substitution.
pdf (552 kb)
tex (31 kb)
References
- Anderson I.M., Kamran N., The variational bicomplex for hyperbolic second-order
scalar partial differential equations in the plane, Duke Math. J
87 (1997), 265-319.
- Bäcklund A.V., Einiges über Curven und Flächen Transformationen,
Lund Universitëts Arsskrift 10 (1874), 1-12.
- Bianchi L., Ricerche sulle superficie elicoidali e sulle superficie a curvatura
costante, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1879),
285-341.
- Darboux G., Leçons sur la théorie générale des surfaces et les
applications géométriques du calcul infinitésimal. II,
Gauthier-Villars, Paris, 1889.
- Drinfel'd V.G., Svinolupov S.I., Sokolov V.V., Classification of fifth-order
evolution equations having an infinite series of conservation laws,
Dokl. Akad. Nauk Ukrain. SSR Ser. A (1985), no. 10, 8-10.
- Goursat E., Leçon sur l'intégration des équations aux dérivées
partielles du second ordre á deux variables indépendantes, I, II,
Hermann, Paris, 1896.
- Khabirov S.V., Infinite-parameter families of solutions of nonlinear
differential equations, Sb. Math. 77 (1994), 303-311.
- Kuznetsova M.N., Laplace transformation and nonlinear hyperbolic equations,
Ufa Math. J. 1 (2009), no. 3, 87-96.
- Kuznetsova M.N., On nonlinear hyperbolic equations related with the
Klein-Gordon equation by differential substitutions, Ufa Math. J.
4 (2012), no. 3, 86-103.
- Liouville J., Sur l'equation aux différences partielles $\partial^2 \log
\lambda /\partial u\partial v \pm \lambda /(aa^2)=0$, J. Math. Pures
Appl. 18 (1853), 71-72.
- Meshkov A.G., Sokolov V.V., Hyperbolic equations with third-order symmetries,
Theoret. Math. Phys. 166 (2011), 43-57.
- Sokolov V.V., On the symmetries of evolution equations, Russian Math. Surveys
43 (1988), no. 5, 165-204.
- Soliman A.A., Abdo H.A., New exact solutions of nonlinear variants of the RLN,
the PHI-four and Boussinesq equations based on modified extended direct
algebraic method, Int. J. Nonlinear Sci. 7 (2009),
274-282, arXiv:1207.5127.
- Startsev S.Ya., Hyperbolic equations admitting differential substitutions,
Theoret. Math. Phys. 127 (2001), 460-470.
- Startsev S.Ya., Laplace invariants of hyperbolic equations linearizable by a
differential substitution, Theoret. Math. Phys. 120 (1999),
1009-1018.
- Svinolupov S.I., Second-order evolution equations with symmetries,
Russian Math. Surveys 40 (1985), no. 5, 241-242.
- Tzitzéica G., Sur une nouvelle classe de surfaces, C. R. Acad. Sci.
144 (1907), 1257-1259.
- Zhiber A.V., Shabat A.B., Klein-Gordon equations with a nontrivial group,
Soviet Phys. Dokl. 24 (1979), 607-609.
- Zhiber A.V., Sokolov V.V., Exactly integrable hyperbolic equations of
Liouville type, Russian Math. Surveys 56 (2001), no. 1,
61-101.
- Zhiber A.V., Sokolov V.V., Startsev S.Ya., Darboux integrable nonlinear
hyperbolic equations, Dokl. Math. 52 (1995), 128-130.
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