Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 090, 37 pages      arXiv:1111.7255      http://dx.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.

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References

  1. 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.
  2. Bäcklund A.V., Einiges über Curven und Flächen Transformationen, Lund Universitëts Arsskrift 10 (1874), 1-12.
  3. Bianchi L., Ricerche sulle superficie elicoidali e sulle superficie a curvatura costante, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1879), 285-341.
  4. Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. II, Gauthier-Villars, Paris, 1889.
  5. 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.
  6. Goursat E., Leçon sur l'intégration des équations aux dérivées partielles du second ordre á deux variables indépendantes, I, II, Hermann, Paris, 1896.
  7. Khabirov S.V., Infinite-parameter families of solutions of nonlinear differential equations, Sb. Math. 77 (1994), 303-311.
  8. Kuznetsova M.N., Laplace transformation and nonlinear hyperbolic equations, Ufa Math. J. 1 (2009), no. 3, 87-96.
  9. 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.
  10. 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.
  11. Meshkov A.G., Sokolov V.V., Hyperbolic equations with third-order symmetries, Theoret. Math. Phys. 166 (2011), 43-57.
  12. Sokolov V.V., On the symmetries of evolution equations, Russian Math. Surveys 43 (1988), no. 5, 165-204.
  13. 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.
  14. Startsev S.Ya., Hyperbolic equations admitting differential substitutions, Theoret. Math. Phys. 127 (2001), 460-470.
  15. Startsev S.Ya., Laplace invariants of hyperbolic equations linearizable by a differential substitution, Theoret. Math. Phys. 120 (1999), 1009-1018.
  16. Svinolupov S.I., Second-order evolution equations with symmetries, Russian Math. Surveys 40 (1985), no. 5, 241-242.
  17. Tzitzéica G., Sur une nouvelle classe de surfaces, C. R. Acad. Sci. 144 (1907), 1257-1259.
  18. Zhiber A.V., Shabat A.B., Klein-Gordon equations with a nontrivial group, Soviet Phys. Dokl. 24 (1979), 607-609.
  19. Zhiber A.V., Sokolov V.V., Exactly integrable hyperbolic equations of Liouville type, Russian Math. Surveys 56 (2001), no. 1, 61-101.
  20. Zhiber A.V., Sokolov V.V., Startsev S.Ya., Darboux integrable nonlinear hyperbolic equations, Dokl. Math. 52 (1995), 128-130.

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