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Astrophysics and Space Science

, Volume 344, Issue 2, pp 281–285 | Cite as

Comments on “Two exact solutions to the general relativistic Binet’s equation”

  • Zenonas Navickas
  • Minvydas Ragulskis
Letter

Abstract

In their recent manuscript He and Zeng claim that they have solved the general relativistic Binet’s orbit equation using the exp-function method and have obtained two exact solutions useful for theoretical analysis. We argue that the obtained solutions do not satisfy the original differential equation. Moreover, we present the alternative framework for the solution of the general relativistic Binet’s orbit equation.

Keywords

Binet’s orbit equation Exp-function method Solitary solution 

Notes

Acknowledgements

Financial support from the Lithuanian Science Council under project No. MIP-041/2011 is acknowledged.

References

  1. Airault, H., McKean, H.P., Moser, J.: Rational and elliptic solutions of the KdV equation and related many-body problems. Commun. Pure Appl. Math. 30, 95–148 (1977) MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. D’Eliseo, M.M.: The first-order orbital equation. Am. J. Phys. 75(4), 352–355 (2007) ADSCrossRefGoogle Scholar
  3. D’Eliseo, M.M.: The gravitational ellipse. J. Math. Phys. 50, 022901 (2009) MathSciNetADSCrossRefGoogle Scholar
  4. Gardner, C.S., Green, J.M., Kruskal, M.D., Miura, R.M.: Method for solving the Korteweg-de Vries equations. Phys. Rev. Lett. 19, 1095–1097 (1967) ADSzbMATHCrossRefGoogle Scholar
  5. He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fractals 30(3), 700–708 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. He, J.-H., Zeng, D.-Q.: Two exact solutions to general relativistic Binet’s equation. Astrophys. Space Sci. 323, 97–98 (2009) ADSzbMATHCrossRefGoogle Scholar
  7. Hirota, R.: Exact solution of Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971) ADSzbMATHCrossRefGoogle Scholar
  8. Korteweg, D.J., de Vries, G.: On the change of the form of long waves advancing in rectangular canal, and on a new type of stationary waves. Philos. Mag. 39, 422–443 (1895) zbMATHGoogle Scholar
  9. Kudryashov, N.A.: Seven common errors in finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 14(9–10), 3507–3529 (2009) MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. Lax, P.D.: Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math. 21, 467–490 (1968) MathSciNetzbMATHCrossRefGoogle Scholar
  11. Navickas, Z., Ragulskis, M.: How far one can go with the Exp function method? Appl. Math. Comput. 211(2), 522–530 (2009) zbMATHCrossRefGoogle Scholar
  12. Navickas, Z., Ragulskis, M., Bikulciene, L.: Be careful with the Exp-function method—additional remarks. Commun. Nonlinear Sci. Numer. Simul. 15(12), 3874–3886 (2010) MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. Navickas, Z., Bikulciene, L., Rahula, M., Ragulskis, M.: Algebraic operator method for the construction of solitary solutions to nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. (2013). doi: 10.1016/j.cnsns.2012.10.009 Google Scholar
  14. Rosales, R.R.: The similarity solution for Korteweg-de Vries equation and related Painleve transcendent. Proc. R. Soc. Lond. A 361, 265–275 (1978) MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. Ryabov, P.N., Chesnokov, S.A.: A note on the “Exp-function method for traveling waves of nonlinear evolution equations”. Appl. Math. Comput. 218(17), 9024–9026 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  16. Saca, J.M.: An exact solution to the relativistic advance of perihelion: correcting the Einstein approximation. Astrophys. Space Sci. 315, 365 (2008) ADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Department of Applied MathematicsKaunas University of TechnologyKaunasLithuania
  2. 2.Research Group for Mathematical and Numerical Analysis of Dynamical SystemsKaunas University of TechnologyKaunasLithuania

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