Communications in Mathematical Physics

, Volume 158, Issue 2, pp 397–430 | Cite as

Nonpersistence of breather families for the perturbed sine Gordon equation

  • Jochen Denzler


We show that, up to one exception and as a consequence of first order perturbation theory only, it is impossible that a large portion of the well-known family of breather solutions to the sine Gordon equation could persist under any nontrivial perturbation of the form
$$u_{tt} - u_{xx} + \sin u = \varepsilon \Delta \left( u \right) + O\left( {\varepsilon ^2 } \right),$$
where δ is an analytic function in anarbitrarily small neighbourhood ofu=0. Improving known results, we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of δ to be small. The single exception is a one-dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in ε.


Neural Network Statistical Physic Complex System Analytic Function Sine 
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  1. 1.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Studies in Appl. Math. Vol. LIII, No. 4, 249–315 (1974)Google Scholar
  2. 2.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Method for solving the sine-Gordon equation. Phys. Rev. Lett.30, 1262–1264 (1973)Google Scholar
  3. 3.
    Barone, A., Esposito, F., Magee, C.J., Scott, A.C.: Theory and applications of the sine-Gordon equation. Rivista del Nuovo Cimento1, 227–267 (1971)Google Scholar
  4. 4.
    Bikbaev, P.F., Kuksin, S.B.: Periodic boundary value problem for the sine-Gordon equation, its small Hamiltonian perturbations, and KAM deformations of finite gap tori. To appear in Algebra i analiz (St. Petersburg Mathematical Journal)3, 3 (1992)Google Scholar
  5. 5.
    Birnir, B., McKean, H., Weinstein, A.: Nonexistence of breathers. Private communication of draft, 1990, to appear in Comm. Pure Appl. Math.Google Scholar
  6. 6.
    Coron, J.-M.: Période minimale pour une corde vibrante de longueur infinie. Comptes Rendues Acad. Sci. Paris A294, 127–129 (1982); erratum A295, 371 (1982)Google Scholar
  7. 7.
    Craig, W., Wayne, C.E.: Newton's method and periodic solutions of nonlinear wave equations. Preprint 1991Google Scholar
  8. 8.
    Craig, W., Wayne, C.E.: Nonlinear waves and the KAM theorem: nonlinear degeneracies. Preprint 1991Google Scholar
  9. 9.
    Delshams, A., Martinez-Seara, M.T.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Preprint 1991Google Scholar
  10. 10.
    Denzler, J.: Nonpersistence of Breather solutions under perturbation of the sine Gordon equation. In: Proceedings on dynamical systems, Euler International Mathematical Institute, Sankt Petersburg, November 1991, ed. V.F. Lazutkin, to appear in Birkhäuser, BaselGoogle Scholar
  11. 11.
    Denzler, J.: Nonpersistence of Breathers for the perturbed sine Gordon equation. PhD thesis number 9954, ETH Zürich, Switzerland, 1992. Copy available from the authorGoogle Scholar
  12. 12.
    Eleonskii, V.M., Kulagin, N.E., Novozhilova, N.S., Silin, V.P.: Asymptotic expansions and qualitative analysis of finite-dimensional models in nonlinear field theory. Teor. i Mat. Fizika60, 3, 395–403 (1984)Google Scholar
  13. 13.
    Forest, M.G., McLaughlin, D.W.: Spectral theory for the periodic sine Gordon equation: a concrete viewpoint. J. Math. Phys.23, 1248–1277 (1982)Google Scholar
  14. 14.
    Holmes, P., Marsden, J., Scheurle, J.: Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. Contemp. Math.81, 213–244 (1988)Google Scholar
  15. 15.
    Kaup, D.J.: Closure of the squared Zakharov-Shabat eigenstates. J. Math. Anal. and Appl.54, 849–864 (1976)Google Scholar
  16. 16.
    Kichenassamy, S.: Breather solutions of the nonlinear wave equation. Comm. Pure Appl. Math.44, 789–818 (1991)Google Scholar
  17. 17.
    Kuksin, S.B.: Quasiperiodic solutions of nearly integrable infinite-dimensional Hamiltonian systems. To appear in Springer Lect. Notes in Math.Google Scholar
  18. 18.
    Lazutkin, V.F., Schachmannski, I.G., Tabanov, M.B.: Splitting of separatrices for standard and semistandard mappings. Physica D40, 235–248 (1989)Google Scholar
  19. 19.
    McLaughlin, D.W.: Four examples of the inverse method as a canonical transformation. J. Math. Phys.16, 96–99 (1975)Google Scholar
  20. 20.
    McLaughlin, D.W., Scott, A.C.: Perturbation analysis of fluxon dynamics. Phys. Rev. A18, 4, 1652–1680 (1978)Google Scholar
  21. 21.
    Marsden, J.E., McCracken, M.: The Hopf bifurcation and its applications. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  22. 22.
    Scheurle, J.: Splitting of separatrices and chaos. Lect. in Appl. Math.26, 561–571 (1990)Google Scholar
  23. 23.
    Segur, H., Kruskal, M.D.: Nonexistence of small amplitude Breather solutions inφ 4 theory. Phys. Rev. Lett.58, 747–750 (1987)Google Scholar
  24. 24.
    Siegel, C.L., Moser, J.K.: Lectures on celestial mechanics. Springer Grundlehren 187, 1971Google Scholar
  25. 25.
    Sigal, I.M.: Non-linear wave and Schrödinger equations. I. Instability of periodic and quasiperiodic solutions. Preprint 1991Google Scholar
  26. 26.
    Takhtadzhyan, L.A., Faddeev, L.D.: Essentially nonlinear one-dimensional model of classical field theory. Teor. i Mat. Fizika21, 2, 160–174 (1974)Google Scholar
  27. 27.
    Weinstein, A.: Periodic nonlinear waves on a half-line. Commun. Math. Phys.99, 385–388 (1985); erratum107, 177 (1986)Google Scholar
  28. 28.
    Wolfram, S.: Mathematica; a system for doing mathematics on a Computer. Addison-WesleyGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Jochen Denzler
    • 1
  1. 1.Mathematisches InstitutLudwig-Maximilians-UniversitätMünchenGermany

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