Solitary solutions to a relativistic two-body problem

  • R. Marcinkevicius
  • Z. Navickas
  • M. Ragulskis
  • T. Telksnys
Original Article


Necessary and sufficient conditions for the existence of solitary solutions to a generalized model of a two-body problem perturbed by small post-Newtonian relativistic term are derived in this paper. It is demonstrated that kink, bright and dark solitary solutions exist in the model, when the relativistic effects are treated as higher order perturbations. Numerical experiments are used to verify theoretical results.


Methods: analytical Waves Gravitation Relativistic processes 



This research was funded by a grant (No. MIP078/15) from the Research Council of Lithuania.


  1. Abouelmagd, E.I., Elshaboury, S.M., Selim, H.H.: Astrophys. Space Sci. 361, 37 (2015) Google Scholar
  2. Akhmediev, N.N., Ankiewicz, A.: Dissipative Solitons: From Optics to Biology and Medicine. Springer, Berlin (2010) zbMATHGoogle Scholar
  3. Anglin, J.: Nat. Phys. 4, 437 (2008) CrossRefGoogle Scholar
  4. Aslan, I., Marinakis, V.: Commun. Theor. Phys. 56, 397 (2011) MathSciNetCrossRefGoogle Scholar
  5. Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Savescu, M., Milovic, D., Khan, K.R., Mahmood, M.F., Jovanoski, Z., Biswas, A.: Optik 125, 4935 (2014) ADSCrossRefGoogle Scholar
  6. Billam, T.P., Weiss, C.: Nat. Phys. 10, 902 (2014) CrossRefGoogle Scholar
  7. Bingham, R., Trines, R., Mendonça, J.T., Silva, L.O., Shukla, P.K., Dunlop, M.W., Vaivads, A., Davies, J.A., Bamford, R.A., Mori, W.B., Tynan, G.: AIP Conf. Proc. 1061, 1 (2008) ADSCrossRefGoogle Scholar
  8. Chen, Z., Segev, M., Christodoulides, D.N.: J. Opt. A, Pure Appl. Opt. 5 (2003) Google Scholar
  9. Dauxois, T., Peyrard, M.: Physics of Solitons. Cambridge University Press, Cambridge (2006) zbMATHGoogle Scholar
  10. D’Eliseo, M.M.: Am. J. Phys. 75, 352 (2007) ADSCrossRefGoogle Scholar
  11. D’Eliseo, M.M.: J. Math. Phys. 022901 (2009) Google Scholar
  12. Donadello, S., Serafini, S., Tylutki, M., Pitaevskii, L.P., Dalfovo, F., Lamporesi, G., Ferrari, G.: Phys. Rev. Lett. 113, 065302 (2014) ADSCrossRefGoogle Scholar
  13. Kivshar, Y., Agraval, G.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, Berlin (2003) Google Scholar
  14. Korteweg, D.J., de Vries, G.: Philos. Mag. Ser. 5 39, 422 (1985) CrossRefGoogle Scholar
  15. Krishnan, E.V., Ghabshi, M.A., Mirzazadeh, M., Bhrawy, A.H., Biswas, A., Belic, M.: J. Comput. Theor. Nanosci. 12, 4809 (2015) Google Scholar
  16. Kudryashov, N.A., Loguinova, N.B.: Commun. Nonlinear Sci. Numer. Simul. 14, 1891 (2009) ADSMathSciNetCrossRefGoogle Scholar
  17. Mbuli, L.M., Maharaj, S.K., Bharuthram, R., Singh, S.V., Lakhina, G.S.: Phys. Plasmas 22, 062307 (2015) ADSCrossRefGoogle Scholar
  18. Mercier, M.J., Mathur, M., Gostiaux, L., Gerkema, T., Magalhaes, J.M., Silva, J.C.B.D., Dauxois, T.: J. Fluid Mech. 704, 37 (2012) ADSCrossRefGoogle Scholar
  19. Navickas, Z., Ragulskis, M.: Astrophys. Space Sci. 344, 281 (2013) ADSCrossRefGoogle Scholar
  20. Navickas, Z., Bikulciene, L., Ragulskis, M.: Appl. Math. Comput. 216, 2380 (2010) MathSciNetGoogle Scholar
  21. Navickas, Z., Bikulciene, L., Rahula, M., Ragulskis, M.: Commun. Nonlinear Sci. Numer. Simul. 18, 1374 (2013) ADSMathSciNetCrossRefGoogle Scholar
  22. Navickas, Z., Ragulskis, M., Telksnys, T.: Appl. Math. Comput. (2016). doi: 10.1016/j.amc.2016.02.049
  23. Polyanin, A.D., Zaitsev, V.F.: Handbook of Exact Solutions for Ordinary Differential Equations. Chapman & Hall/CRC Press, London/Boca Raton (2003) zbMATHGoogle Scholar
  24. Remoissenet, M.: Waves Called Solitons: Concepts and Experiments. Springer, Berlin (1999) CrossRefzbMATHGoogle Scholar
  25. Saca, J.M.: Astrophys. Space Sci. 315, 365 (2008) ADSCrossRefGoogle Scholar
  26. Savescu, M., Bhrawy, A.H., Alshaery, A.A., Hilal, E.M., Khan, K.R., Mahmood, M.F., Biswas, A.: J. Mod. Opt. 61, 441 (2014) ADSMathSciNetCrossRefGoogle Scholar
  27. Scott, A. (ed.): Encyclopedia of Nonlinear Science. Routledge, New York (2004) zbMATHGoogle Scholar
  28. Telksnys, T., Navickas, Z., Ragulskis, M.: In: ICCAM 2016: International Conference on Computational and Applied Mathematics, Dubai, UAE, Jan. 28–29, vol. 18, p. 4305 (2016) Google Scholar
  29. To, F.T., Fung, P.C.W., Au, C.: Astrophys. Space Sci. 184, 313 (1991) ADSMathSciNetCrossRefGoogle Scholar
  30. Trines, R., Bingham, R., Dunlop, M.W., Vaivads, A., Davies, J.A., Mendonça, J.T., Silva, L.O., Shukla, P.K.: Phys. Rev. Lett. 99, 205006 (2007) ADSCrossRefGoogle Scholar
  31. Vitanov, N.K., Dimitrova, Z.I.: Commun. Nonlinear Sci. Numer. Simul. 15, 2836 (2010) ADSMathSciNetCrossRefGoogle Scholar
  32. Vitanov, N.K., Jordanov, I.P., Dimitrova, Z.I.: Appl. Math. Comput. 215, 2950 (2009) MathSciNetGoogle Scholar
  33. Yakhshiev, U.T.: Phys. Lett. B 749, 507 (2015) ADSCrossRefGoogle Scholar
  34. Yang, J.: Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, Philadelphia (2010) CrossRefzbMATHGoogle Scholar
  35. Zhou, Q., Zhu, Q., Yu, H., Liu, Y., Wei, C., Yao, P., Bhrawy, A.H., Biswas, A.: Laser Phys. 25, 025402 (2015) ADSCrossRefGoogle Scholar
  36. Zhou, Q., Zhong, Y., Mirzazadeh, M., Bhrawy, A.H., Zerrad, E., Biswas, A.: Wave Random Complex 26, 204 (2016) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • R. Marcinkevicius
    • 1
  • Z. Navickas
    • 2
  • M. Ragulskis
    • 3
  • T. Telksnys
    • 3
  1. 1.Department of Software EngineeringKaunas University of TechnologyKaunasLithuania
  2. 2.Department of Mathematical ModelingKaunas University of TechnologyKaunasLithuania
  3. 3.Research Group for Mathematical and Numerical Analysis of Dynamical SystemsKaunas University of TechnologyKaunasLithuania

Personalised recommendations