International Journal of Theoretical Physics

, Volume 45, Issue 1, pp 53–63 | Cite as

Internal Modes of Relativistic Solitons

  • A. R. Gharaati
  • N. Riazi
  • and F. Mohebbi


We apply a linear perturbation analysis to investigate the relationship between soliton oscillations and the integrability of nonlinear PDEs in bi-dimensional spacetime. For this purpose, we consider a localized solution of the nonlinear differential equation, and study small amplitude fluctuations around it. The linearized equation is a Schrödinger-like, eigenvalue problem. By considering several nonlinear PDEs, which are known to have soliton and solitary wave solutions, we find that in systems which are integrable, this eigenvalue equation has one and only one bound state with zero frequency. Non-integrable equations—in contrast—show extra bound states. The time evolution of the oscillations are also calculated, using a numerical program to integrate the time-dependent equation. The behavior of the modes are studied, using the Fourier transform of the evolving solutions.

Key Words

nonlinear PDEs solitons integrability 


  1. Berry, M. V. (1978). AIP Proceedings 46, 16.MathSciNetCrossRefADSGoogle Scholar
  2. Das, A. (1989). Integrable Models, World Scientific.Google Scholar
  3. Drazin, P. G. and Johnson, R. S. (1989). Solitons: An Introduction, Cambridge University Press, Cambridge.MATHGoogle Scholar
  4. Hasegawa, A. and Kodama, Y. (1995). Solitons in Optical Communications, Oxford University Press.Google Scholar
  5. Ince, E. L. (1927). Ordinary Differential Equations, Longman, London.MATHGoogle Scholar
  6. Khare, A., Habib, S., and Saxena, A. (1997). Physical Review Letters 79(20), 3797.MATHCrossRefADSMathSciNetGoogle Scholar
  7. Lamb, G. L., Jr. (1980). Elements of Soliton Theory, John Wiley and Sons, New York, U.S.A.MATHGoogle Scholar
  8. Lax, P. D. (1968). Communications on Pure and Applied Mathematics 21, 467.MATHCrossRefMathSciNetGoogle Scholar
  9. Pelinovsky, D. E., Kivshar, Y. S., and Afanasjev, V. V. (1998). Physica D 116, 121.MATHCrossRefADSMathSciNetGoogle Scholar
  10. Rajaraman, R. (1982). Solitons and Instantons, North Holland, Elsevier, Amsterdam.MATHGoogle Scholar
  11. Ramani, A., Grammaticos, B., and Bountis, T. (1989). Physics Reports 180(3), 159–245.CrossRefADSMathSciNetGoogle Scholar
  12. Riazi, N. (1993). International Journal of Theoretical Physics 32(11), 2155.CrossRefADSMathSciNetGoogle Scholar
  13. Riazi, N. and Gharaati, A. R. (1998). International Journal of Theoretical Physics 37(3), 1081.MATHCrossRefMathSciNetGoogle Scholar
  14. Turitsyn, S. K. (1993). Physical Review E47, R796.ADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Payame Noor UniversityShirazIran
  2. 2.Physics Department and Biruni ObservatoryShiraz UniversityShirazIran
  3. 3.Physics Department and Biruni ObservatoryShiraz UniversityShirazIran

Personalised recommendations