Nonlinear Oscillations

, Volume 8, Issue 4, pp 513–525 | Cite as

Soliton-like excitations in the one-dimensional electrical transmission line

  • F. B. Pelap
  • M. M. Faye


The dynamics of modulated waves are studied in the one-dimensional discrete nonlinear electrical transmission line. The contribution of the linear dispersive capacitance is taken into account, and it is shown via the reductive perturbation method that the evolution of such waves in this system is governed by the higher-order nonlinear Schrödinger equation. Passing through the Stokes analysis, we establish a generalized criterion for the Benjamin-Feir instability in the network and determine the exact solutions of the obtained wave equation by using the Pathria-Morris approach.


Differential Equation Exact Solution Partial Differential Equation Ordinary Differential Equation Wave Equation 
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  1. 1.
    S. E. Trullinger, V. E. Zakharov, and V. L. Pokrovsky, Solitons, North-Holland, Amsterdam (1986).Google Scholar
  2. 2.
    W. Eckhaus, Studies in the Nonlinear Stability Theory, Springer, Berlin (1965).Google Scholar
  3. 3.
    Y. S. Kivshar and M. Salerno, “Modulational instability in the discrete deformable nonlinear Schrödinger equation,” Phys. Rev. E, 49, 3543–3546 (1994).MathSciNetGoogle Scholar
  4. 4.
    M. Toda, “Theory of nonlinear lattice,” J. Phys. Soc. Jpn., 22, 413 (1967).Google Scholar
  5. 5.
    J. Parkes, “The modulation of weakly nonlinear dispersive waves near the marginal state of instability,” J. Phys. A: Math. Gen., 20, 2025–2036 (1987).Google Scholar
  6. 6.
    F. B. Pelap, T. C. Kofané, N. Flytzanis, and M. Remoissenet, “Waves modulations in the biinductance transmission line,” J. Phys. Soc. Jpn., 70, 2568–2577 (2001).CrossRefGoogle Scholar
  7. 7.
    T. C. Kofané, B. Michaux, and M. Remoissenet, “Theoretical and experimental studies of diatomic lattice solitons using an electrical transmission line,” J. Phys. C, 21, 1395–1412 (1988).Google Scholar
  8. 8.
    M. Remoissenet, Waves Called Solitons, Springer, Berlin (1999).Google Scholar
  9. 9.
    P. Marquie, J. M. Bilbault, and M. Remoissenet, “Generation of envelope and hole solitons in an experimental transmission line,” Phys. Rev. E, 46, 828–835 (1994).Google Scholar
  10. 10.
    T. Taniuti and N. Yajima, “Perturbation method for nonlinear equations,” J. Math. Phys., 10, 1369–1372 (1969).CrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Ketchakeu, T. C. Kofané, and A. Zibi, “Modulational instability in an extended Φ 4 model near the marginal state,” Phys. Scr., 44, 505–510 (1991).Google Scholar
  12. 12.
    F. B. Pelap and T. C. Kofané, “The revised modulational instability criterion: I. The monoinductance transmission line,” Phys. Scr., 57, 410–415 (1998).CrossRefGoogle Scholar
  13. 13.
    T. Kakutani and K. Michihiro, “Marginal state of modulational instability—note of the Benjamin-Feir instability,” J. Phys. Soc. Jpn., 52, 4129–4137 (1983).CrossRefGoogle Scholar
  14. 14.
    T. Kawahara, J. Sakai, and T. Kakutani, “Nonlinear wave modulation in dispersive media,” J. Phys. Soc. Jpn., 29, 1068–1073 (1970).Google Scholar
  15. 15.
    R. Ndohi and T. C. Kofané, “Solitary waves in ferromagnetic chains near marginal state of instability,” Phys. Lett. A, 154, 377–379 (1991).CrossRefGoogle Scholar
  16. 16.
    R. S. Johnson, Proc. Roy. Soc. A, 357, 131 (1977).MATHGoogle Scholar
  17. 17.
    R. J. Deissler, J. Stat. Phys., 54, 1459 (1989).CrossRefMathSciNetGoogle Scholar
  18. 18.
    Y. Pomeau, “Front motion, metastability and structural bifurcations in hydrodynamics,” Physica D, 23, 3–11 (1986).CrossRefGoogle Scholar
  19. 19.
    T. B. Benjamin and J. F. Feir, “The disintegration of wave trains on deep water,” J. Fluid Mech., 27, 417–430 (1967).CrossRefGoogle Scholar
  20. 20.
    J. T. Stuart and R. C. Di Prima, “The Eckhauss and Benjamin-Feir resonance mechanisms,” Phys. Roy. Soc. London. Ser. A, 362, 27–41 (1978).Google Scholar
  21. 21.
    R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London (1987).Google Scholar
  22. 22.
    F. B. Pelap and T. C. Kofané, “Modulational instability in some physical systems,” Phys. Scr., 64, 410–412 (2001).CrossRefGoogle Scholar
  23. 23.
    E. Kengne, “Modified Ginzburg-Landau equation and Benjamin-Feir instability,” Nonlin. Oscillations, 6, No. 3, 346–356 (2003).MATHMathSciNetGoogle Scholar
  24. 24.
    D. Pathria and J. L. Morris, “Exact solutions for a generalized nonlinear Schrödinger equation,” Phys. Scr., 39, 673–679 (1989).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • F. B. Pelap
    • 1
  • M. M. Faye
    • 2
  1. 1.Université de DschangDschangCameroon
  2. 2.Université Cheikh Anta DiopDakarSenegal

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