The European Physical Journal Special Topics

, Volume 225, Issue 6–7, pp 943–958 | Cite as

Nonlinear wave propagation in discrete and continuous systems

  • V.M. Rothos
Review Session A: Reviews
Part of the following topical collections:
  1. Mathematical Modeling of Complex Systems


In this review we try to capture some of the recent excitement induced by a large volume of theoretical and computational studies addressing nonlinear Schrödinger models (discrete and continuous) and the localized structures that they support. We focus on some prototypical structures, namely the breather solutions and solitary waves. In particular, we investigate the bifurcation of travelling wave solution in Discrete NLS system applying dynamical systems methods. Next, we examine the combined effects of cubic and quintic terms of the long range type in the dynamics of a double well potential. The relevant bifurcations, the stability of the branches and their dynamical implications are examined both in the reduced (ODE) and in the full (PDE) setting. We also offer an outlook on interesting possibilities for future work on this theme.


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  1. 1.
    M. Albiez, R. Gati, J. Folling, S. Hunsmann, M. Cristiani, M.K. Oberthaler, Phys. Rev. Lett. 95, 010402 (2005)ADSCrossRefGoogle Scholar
  2. 2.
    M.J. Ablowitz, J.F. Ladik, J. Math. Phys. 17, 1011 (1976)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    M.J. Ablowitz, Z.H. Musslimani, G. Biondini, Phys. Rev. E 65, 026602 (2002)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    A.A. Aigner, A.R. Champneys, V.M. Rothos, Physica D 186, 148 (2003)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    E.J. Doedel, A.R. Champneys, T.R. Fairgrieve, Y.A. Kuznetsov, B. Sandstede, X.J. Wang. Auto97 continuation and bifurcation software for ordinary differential equations. (1997)
  6. 6.
    D.B. Duncan, J.C. Eilbeck, H. Feddersen, J.A.D. Wattis, Physica D 68, 1 (1993)ADSCrossRefGoogle Scholar
  7. 7.
    J.C. Eilbeck, R. Flesch, Phys. Lett. A 149, 200 (1990)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    P.Tsilifis, P.G. Kevrekidis, V. Rothos, JPHYS A 47, 035201 (2014)ADSGoogle Scholar
  9. 9.
    S. Flach, K. Kladko, Physica D 127, 61 (1999)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    T.R.O. Melvin, A.R. Champneys, P.G. Kevrekedis, J. Cuevas, Phys. Rev. Lett. 97 Art.No. 124101 (2006)ADSCrossRefGoogle Scholar
  11. 11.
    T.R.O. Melvin, A.R. Champneys, P.G. Kevrekedis, J. Cuevas, Physica D 237, 551 (2008)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    T.R.O. Melvin, A.R. Champneys, D.E. Pelinovsky (preprint) (University of Bristol, 2008)Google Scholar
  13. 13.
    O.F. Oxtoby, I.V. Barashenkov, Phys. Rev. E 76 036603 (2007)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    D.E. Pelinovsky, Nonlinearity 19, 2695 (2006)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    D.E. Pelinovsky, T.R.O. Melvin, A.R. Champneys, Physica D 236, 22 (2007)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    D.E. Pelinovsky, V.M. Rothos, Physica D 202, 16 (2005)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    D.E. Pelinovsky, J. Yang, Chaos 15, 037115 (2005)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    M.J.D. Powell, A hybrid method for nonlinear algebraic equations, Numerical Methods for Nonlinear Algebraic Equations. (Gordon and Breach, 1970)Google Scholar
  19. 19.
    M. Remoissenet, M. Peyrard (eds.) Nonlinear Coherent Structures in Physics and Biology (Springer-Verlag Berlin, 1991)Google Scholar
  20. 20.
    M. Salerno, Phys. Rev. A 46, 6856 (1992)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    A.V. Savin, Y. Zolotaryuk, J.C. Eilbeck, Phys. D. 138, 267 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    D.N. Christodoulides, F. Lederer, Y. Silderberg, Nature 424, 817 (2003)ADSCrossRefGoogle Scholar
  23. 23.
    J.W. Fleischer, T. Carmon, M. Segev, N.K. Efremidis, D.N. Christodoulides, Phys. Rev. Lett. 90, 023902 (2003)ADSCrossRefGoogle Scholar
  24. 24.
    P.G. Kevrekidis, K.O. Rasmussen, A.R. Bishop, Int. J. Mod. Phys. B 15, 2833 (2001)ADSCrossRefGoogle Scholar
  25. 25.
    C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, 2002)Google Scholar
  26. 26.
    L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, 2003)Google Scholar
  27. 27.
    J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 2008)Google Scholar
  28. 28.
    Yu.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)Google Scholar
  29. 29.
    J. Yang, Phys. Rev. Lett. 91, 143903 (2003)ADSCrossRefGoogle Scholar
  30. 30.
    C. Cambournac, T. Sylvestre, H. Maillotte, B. Vanderlinden, P. Kockaert, Ph. Emplit, M. Haelterman, Phys. Rev. Lett. 89, 083901 (2002)ADSCrossRefGoogle Scholar
  31. 31.
    P.G. Kevrekidis, Z. Chen, B.A. Malomed, D.J. Frantzeskakis, M.I. Weinstein, Phys. Lett. A 340, 275 (2005)ADSCrossRefGoogle Scholar
  32. 32.
    M. Ornigotti, G. Della Valle, D. Gatti, S. Longhi, Phys. Rev. A 76, 023833 (2007)ADSCrossRefGoogle Scholar
  33. 33.
    J. Yang, T.R. Akylas, Stud. Appl. Math. 111, 359 (2003)MathSciNetCrossRefGoogle Scholar
  34. 34.
    C. Wang, P.G. Kevrekidis, D.J. Frantzeskakis, B.A. Malomed, Physica D 240, 805 (2011)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    P. Engels, C. Atherton Phys. Rev. Lett. 99, 160405 (2007)ADSCrossRefGoogle Scholar

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© EDP Sciences and Springer 2016

Authors and Affiliations

  1. 1.Department of Mechanical Engineering and Laboratory of Nonlinear Mathematicsand Aristotle University of ThessalonikiThessalonikiGreece

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