Skip to main content
SpringerLink
Log in
Book cover

Integrable Systems and Applications pp 328–335Cite as

On the linear stability of solitary waves in Hamiltonian systems with symmetry

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Physics ((LNP,volume 342))

Abstract

For a class of nonlinear Schrödinger equations we saw that there is a one to one correspondence between orbital, energetic and linear stability of solitary waves provided d″ (ω) is nonzero. The case d″ (ω) = 0 is critical in the sense that this equivalence breaks down. This result will also extend to the abstract theory provided the linearized operator satisfies the assumptions on its spectrum listed in the introduction. In addition, a more detailed analysis of the critical case will show that for linear stability a conditional stability. result can be obtained which corresponds to the (nonlinear) stability results by M. Weinstein [11].

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D.D. Holm, J.E. Marsden, T. Ratin and A. Weinstein, Phys. Rep. 123, 1 (1985)

    Google Scholar 

  2. J. Stubbe and L. Vazquez, to appear in “Mathematics + Physics. Lectures on recent results” Vol. 3, L. Streit, editor, World Scientific, Singapore

    Google Scholar 

  3. C.L. Siegel and J.K. Moser, “Lectures on Celestial Mechanics”, Springer 1971

    Google Scholar 

  4. M. Grillakis, J. Shatah and W. Strauss, J. Funct. Anal. 74, 160 (1987)

    Google Scholar 

  5. J. Stubbe, Portugagaliae Mathematica, to appear

    Google Scholar 

  6. D.W. McLaughlin and A.C. Scott, Phys. Rev. A18, 1652 (1978)

    Google Scholar 

  7. M. Weinstein, Siam J. Math. Anal. Vol. 16, 472 (1985)

    Google Scholar 

  8. J. Stubbe, to appear in the Proceedings of the IVth German-French meeting on Mathematical Physics, Marseille 1988

    Google Scholar 

  9. H. Berestycki and P.L. Lions, Arch. Rat. Mech. Anal. 82, 313 (1983)

    Google Scholar 

  10. M. Weinstein, Comm. Math. Phys. 87, 567 (1983)

    Google Scholar 

  11. M. Weinstein, Comm. Part. Diff. Eq. 11, 545 (1986)

    Google Scholar 

  12. J. Shatah, Comm. Math. Phys. 91, 313 (1983)

    Google Scholar 

Download references

Author information

Author notes

  1. Joachim Stubbe

    Present address: Fakultät für Physik, Universität Bielefed, Postfach 8640, D-4800, Bielefed 1, Germany

Authors and Affiliations

  1. Département de Physiaue Théorique, Université de Genève, 24, quai Ernest-Anserment, CH-1211, Genève 4, Suisse

    Joachim Stubbe

Authors
  1. Joachim Stubbe
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

M. Balabane P. Lochak C. Sulem

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag

About this paper

Cite this paper

Stubbe, J. (1989). On the linear stability of solitary waves in Hamiltonian systems with symmetry. In: Balabane, M., Lochak, P., Sulem, C. (eds) Integrable Systems and Applications. Lecture Notes in Physics, vol 342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0035680

Download citation

  • DOI: https://doi.org/10.1007/BFb0035680

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51615-6

  • Online ISBN: 978-3-540-46714-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation