Existence and stability of driven and damped phase-locked NLS-solitons and SG-breathers

  • K. H. Spatschek
  • M. Taki
  • Th. Eickermann
Part III. Nonlinear Wave Propagation: Numerical and Theoretical Studies
Part of the Lecture Notes in Physics book series (LNP, volume 353)


The spatio-temporal behaviors of possible solutions of driven and damped nonlinear Schrödinger and sine-Gordon equations are analyzed. In the first part, the nonlinear Schrödinger equation is considered also as the limiting case corresponding to weak driving and damping in the sine-Gordon equation, with driving frequency close to one. It is shown that in this limit a phase-locked breather with m/n = 1/1 exists and undergoes period-doubling bifurcations. The close analogy to the dynamics of a phase-locked Schrödinger soliton is demonstrated in detail. In the second part, for smaller driving frequencies, the existence and stability of subharmonically phase-locked breathers is discussed. The quasi-periodic route to chaos is re-investigated and special attention is given to the length-dependence of the results. The findings are compared with predictions of simpler models in nonlinear dynamics. The whole paper consists of analytical and numerical results. For the latter, improved diagnostic tools have been developed.


Hopf Bifurcation Discrete Eigenvalue Inverse Scattering Transform Driver Amplitude Spectral Diagnostics 
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  1. 1.
    M.J. Ablowitz and H. Segur, Solitons and the Inverse scattering Transform (Siam, Philadelphia 1981).Google Scholar
  2. 2.
    K.H. Spatschek, H. Pietsch, E.W. Laedke, and. Th. Eickermann, in Singular Behavior and Nonlinear Dynamics, eds. St.Pnevmatikos, T. Bountis, and Sp. Pnevmatikos (World Scientific, Singapore 1989), Vo1.2, p.555.Google Scholar
  3. 3.
    G. Iooss, Bifurcation of Maps and Applications (Mathematical Studies, Vol. 36, North-Holland, Amsterdam 1979)Google Scholar
  4. 4.
    J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Heidelberg 1986).Google Scholar
  5. 5.
    A.R. Bishop, M.G. Forest, D.W. McLaughlin, and E.A. Overman II, Physica D 23, 293 (1986).CrossRefGoogle Scholar
  6. 6.
    M. Maki and K.H. Spatschek, J. de Physique C3, Tome 50, 77 (1989).Google Scholar
  7. 7.
    M. Taki, K.H. Spatschek, J.C. Fernandez, R. Grauer, and G. Reinisch, Physica D (1989), in press.Google Scholar
  8. 8.
    K. Nozaki and N. Bekki, Physica D 21, 381 (1986).CrossRefGoogle Scholar
  9. 9.
    M.G. Forest and D.W. McLaughlin, J. Math. Phys. 23, 1248 (1982).CrossRefGoogle Scholar
  10. 10.
    E.R. Tracey and H.H. Chen, Phys. Rev. A 37, 815 (1988).CrossRefPubMedGoogle Scholar
  11. 11.
    P. Grassberger and I. Procaccia, Phys. Rev. Lett. 50, 346 (1983).CrossRefGoogle Scholar
  12. 12.
    A. Mazor and A.R. Bishop, Physica D 27, 269 (1987).CrossRefGoogle Scholar
  13. 13.
    M.H. Jensen, P. Bak, and T. Bohr, Phys.Rev. A 30, 1960 (1984) ibid. 1970(1984).CrossRefGoogle Scholar
  14. 14.
    D.G. Aronson, M.A. Chory, G.R. Hall, and R.P. McGehee, Commun. Math. Phys. 83,303 (1982).CrossRefGoogle Scholar
  15. 15.
    S. Newhouse, D. Ruelle, and F. Takens, Commun. Math. Phys. 64, 35 (1978)CrossRefGoogle Scholar
  16. 16.
    H. Haucke and R. Ecke, Physica D 25, 307 (1987).CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • K. H. Spatschek
    • 1
  • M. Taki
    • 1
  • Th. Eickermann
    • 1
  1. 1.Institut für Theoretische Physik IHeinrich-Heine-Universität DüsseldorfDüsseldorfGermany

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