Skip to main content
SpringerLink
Log in

Stability Analysis of Pulses via the Evans Function: Dissipative Systems

  • Chapter
  • First Online:
Dissipative Solitons

Part of the book series: Lecture Notes in Physics ((LNP,volume 661))

Abstract

Linear stability analysis of pulses is considered in this review chapter. The Evans function is an analytic tool whose zeros correspond to eigenvalues. Herein, the general manner of its construction shown. Furthermore, the construction is done explicitly for the linearization of the nonlinear Schrödinger equation about the 1-soliton solution. In an explicit calculation, it is shown how the Evans function can be used to track the non-zero eigenvalues arising from a dissipative perturbation of the nonlinear Schrödinger equation which arises in the context of pulse propagation in nonlinear optical fibers.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. P. Fife and J. McLeod, Arch. Rat. Mech. Anal., 65, 335, (1977).

    Google Scholar 

  2. J. Evans, Indiana U. Math. J., 21, 877, (1972).

    Google Scholar 

  3. J. Evans, Indiana U. Math. J., 22, 75, (1972).

    Google Scholar 

  4. J. Evans, Indiana U. Math. J., 22, 577, (1972).

    Google Scholar 

  5. J. Evans, Indiana U. Math. J., 24, 1169, (1975).

    Google Scholar 

  6. C.K.R.T. Jones, Trans. AMS, 286, 431, (1984).

    Google Scholar 

  7. J. Alexander, R. Gardner and C.K.R.T. Jones, J. Reine Angew. Math., 410, 167, 1990

    MATH  Google Scholar 

  8. T. Kapitula and B. Sandstede, Physica D, 124, 58, (1998).

    Google Scholar 

  9. T. Kapitula and B. Sandstede, (to appear in Disc. Cont. Dyn. Sys.), (2003).

    Google Scholar 

  10. M. Ablowitz and P. Clarkson, Solitons, Nonlinear Evolution Equations, and Inverse Scattering, (London Math. Soc. Lecture Note Series, 149, Cambridge U. Press, 1991)

    Google Scholar 

  11. M. Ablowitz, D. Kaup, A. Newell and H. Segur, Stud. Appl. Math., 53, 249, (1974)

    Google Scholar 

  12. J. Alexander and C.K.R.T. Jones, J. Reine Angew. Math., 446, 49, (1994).

    Google Scholar 

  13. J. Alexander and C.K.R.T. Jones, Z. Angew. Math. Phys., 44, 189, (1993).

    Google Scholar 

  14. A. Bose and C.K.R.T. Jones, Indiana U. Math. J., 44, 189, (1995).

    Google Scholar 

  15. T. Bridges and G. Derks, Proc. Royal Soc. London A, 455, 2427, (1999).

    Google Scholar 

  16. T. Bridges and G. Derks, Arch. Rat. Mech. Anal., 156 1, (2001).

    Google Scholar 

  17. R. Gardner and K. Zumbrun, Comm. Pure Appl. Math., 51, 797, (1998).

    Google Scholar 

  18. T. Kapitula, Physica D, 116, 95, (1998).

    Google Scholar 

  19. R. Pego and M. Weinstein, Phil. Trans. R. Soc. Lond. A, 340, 47, (1992).

    Google Scholar 

  20. R. Pego and M. Weinstein, Evans’’ function, and Melnikov’s integral, and solitary wave instabilities, In: Differential Equations with Applications to Mathematical Physics, (Academic Press, Boston, 1993), pp.273–286

    Google Scholar 

  21. T. Kato, Perturbation Theory for Linear Operators, (Springer-Verlag, Berlin, 1980).

    Google Scholar 

  22. J. Swinton, Phys. Lett. A, 163, 57, (1992).

    Google Scholar 

  23. T. Bridges and G. Derks, Phys. Lett. A, 251, 363, (1999).

    Google Scholar 

  24. T. Bridges, Phys. Rev. Lett., 84, 2614, (2000).

    Google Scholar 

  25. B. Sandstede, Handbook of Dynamical Systems, (Elsevier Science, North-Holland, 2002), Vol. 2, Chapter 18, pp. 983–1055.

    Google Scholar 

  26. T. Kapitula, J.N. Kutz and B. Sandstede, Indiana U. Math. J., 53, 1095, (2004).

    Google Scholar 

  27. W.A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics 629, (Springer-Verlag, Berlin, 1978).

    Google Scholar 

  28. D. Peterhof, B. Sandstede and A. Scheel, J. Diff. Eq., 140, 266, (1997).

    Google Scholar 

  29. T. Kapitula and B. Sandstede, SIAM J. Math. Anal., 33, 1117, (2002).

    Google Scholar 

  30. D. Kaup, SIAM J. Appl. Math., 31, 121, (1976).

    Google Scholar 

  31. D. Kaup, Phys. Rev. A, 42, 5689, (1990).

    Google Scholar 

  32. D. Kaup, J. Math. Anal. Appl., 54, 849, (1976).

    Google Scholar 

  33. J. Yang, J. Math. Phys., 41, 6614, (2000).

    Google Scholar 

  34. J. Yang, Phys. Lett. A, 279, 341, (2001).

    Google Scholar 

  35. T. Kapitula, P. Kevrekidis and B. Sandstede, Physica D, 195, 263, (2004).

    Google Scholar 

  36. T. Kapitula, SIAM J. Math. Anal., 30, 273, (1999).

    Google Scholar 

  37. Y. Kivshar, D. Pelinovsky, T. Cretegny and M. Peyrard, Phys. Rev. Lett., 80, 5032, (1998).

    Google Scholar 

  38. D. Pelinovsky, Y. Kivshar and V. Afanasjev, Physica D, 116, 121, (1998)

    Google Scholar 

  39. W. Van Saarloos and P. Hohenberg, Physica D, 560, 303, (1992).

    Google Scholar 

  40. Y. Kodama, M. Romagnoli and S. Wabnitz, Elect. Lett., 28,1981, (1992).

    Google Scholar 

  41. T. Kapitula and B. Sandstede, J. Opt. Soc. Am. B, 15, 2757, (1998).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, NM 87131, Albuquerque, USA

    T. Kapitula

Authors
  1. T. Kapitula
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Nail Akhmediev Adrian Ankiewicz

Rights and permissions

Reprints and permissions

About this chapter

Cite this chapter

Kapitula, T. Stability Analysis of Pulses via the Evans Function: Dissipative Systems. In: Akhmediev, N., Ankiewicz, A. (eds) Dissipative Solitons. Lecture Notes in Physics, vol 661. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10928028_16

Download citation

Publish with us

Policies and ethics

Search

Navigation