Skip to main content
SpringerLink
Log in

On Asymptotic Stability of Kink for Relativistic Ginzburg–Landau Equations

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We prove the asymptotic stability of kink for the nonlinear relativistic wave equations of the Ginzburg–Landau type in one space dimension: for any odd initial condition in a small neighborhood of the kink, the solution, asymptotically in time, is the sum of the kink and dispersive part described by the free Klein–Gordon equation. The remainder converges to zero in a global norm.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Buslaev V.S., Perelman G.S.: Scattering for the nonlinear Schrödinger equations: states close to a soliton. St. Petersburg Math. J. 4(6), 1111–1142 (1993)

    MathSciNet  Google Scholar 

  2. Buslaev V.S., Perelman G.S.: On the stability of solitary waves for nonlinear Schrödinger equations. Am. Math. Soc. Trans. 164(2), 75–98 (1995)

    MATH  Google Scholar 

  3. Buslaev V.S., Sulem C.: On asymptotic stability of solitary waves for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20(3), 419–475 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Cuccagna S.: Stabilization of solutions to nonlinear Schrödinger equations. Commun. Pure Appl. Math. 54, 1110–1145 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cuccagna S.: On asymptotic stability of ground states of NLS. Rev. Math. Phys. 15, 877–903 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cuccagna S., Mizumachi T.: On asymptotic stability in energy space of ground states for nonlinear Schrödinger equations. Commun. Math. Phys. 284(1), 51–77 (2008)

    Article  ADS  MATH  Google Scholar 

  7. Cuccagna S.: On asymptotic stability in 3D of kinks for the \({\phi^4}\) model. Trans. AMS 360(5), 2581–2614 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deift, P.A., Its, A.R., Zhou, X.: Long-time asymptotics for integrable nonlinear wave equations. Important Developments in Soliton Theory (Eds. Fokas A.S. and Zakharov V.E.). Springer, Berlin, 181–204, 1993

  9. Faddeev L.D., Takhtadzhyan L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  10. Gol’dman I.I., Krivchenkov V.D., Kogan V.I., Galitskii V.M.: Problems in Quantum Mechanics. Infosearch LTD., London (1960)

    Google Scholar 

  11. Henry D.B., Perez J.F., Wreszinski W.F.: Stability theory for solitary-wave solutions of scalar field equations. Commun. Math. Phys. 85, 351–361 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Imaikin V., Komech A.I., Vainberg B.: On scattering of solitons for the Klein– Gordon equation coupled to a particle. Commun. Math. Phys. 268(2), 321–367 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Kirr E., Zarnesku A.: On the asymptotic stability of bound states in 2D cubic Schrödinger equation. Commun. Math. Phys. 272(2), 443–468 (2007)

    Article  ADS  MATH  Google Scholar 

  14. Komech A., Kopylova E.: Weighted energy decay for 1D Klein–Gordon equation. Commun. PDE 35(2), 353–374 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kopylov, S.A.: Private communication

  16. Lions, J.L.: Quelques Mèthodes de Rèsolution des Problémes aux Limites non Linéaires. Paris, Dunod, 1969

  17. Merkli M., Sigal I.M.: A time-dependent theory of quantum resonances. Commun. Math. Phys. 201(3), 549–576 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Miller J., Weinstein M.: Asymptotic stability of solitary waves for the regularized long-wave equation. Commun. Pure Appl. Math. 49(4), 399–441 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pego R.L., Weinstein M.I.: Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Pillet C.A., Wayne C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Differ. Equ. 141(2), 310–326 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Reed, M.: Abstract Non-Linear Wave Equations. Lecture Notes in Mathematics, Vol. 507. Springer, Berlin, 1976

  22. Reed M., Simon B.: Methods of Modern Mathematical Physics, Vol. III. Academic Press, New York (1979)

    MATH  Google Scholar 

  23. Rodnianski I., Schlag W., Soffer A.: Dispersive analysis of charge transfer models. Commun. Pure Appl. Math. 58(2), 149–216 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sigal I.M.: Nonlinear wave and Schrödinger equations. I: Instability of periodic and quasiperiodic solutions. Commun. Math. Phys. 153(2), 297–320 (1993)

    Article  ADS  MATH  Google Scholar 

  25. Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. Commun. Math. Phys. 133, 119–146 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Soffer A., Weinstein M.I.: Multichannel nonlinear scattering for nonintegrable equations. II. The case of anisotropic potentials and data. J. Differ. Equ. 98(2), 376–390 (1992)

    Article  MATH  Google Scholar 

  27. Soffer A., Weinstein M.I.: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. Invent. Math. 136, 9–74 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Soffer A., Weinstein M.I.: Selection of the ground states for NLS equations. Rev. Math. Phys. 16(8), 977–1071 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Strauss, W.A.: Nonlinear invariant wave equations. Lecture Notes in Physics, Vol. 73. Springer, Berlin, 197–249, 1978

  30. Tsai T.-P., Yau H.-T.: Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions. Commun. Pure Appl. Math. 55(2), 153–216 (2002)

    Article  MATH  Google Scholar 

  31. Tsai T.-P.: Asymptotic dynamics of nonlinear Schrödinger equations with many bound states. J. Differ. Equ. 192(1), 225–282 (2003)

    Article  ADS  MATH  Google Scholar 

  32. Weinstein M.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16(3), 472–491 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Information Transmission Problems RAS, B. Karetnyi 19, GSP-4, Moscow, 101447, Russia

    E. Kopylova & A. I. Komech

  2. Faculty of Mathematics, Vienna University, Vienna, Austria

    E. Kopylova & A. I. Komech

Authors
  1. E. Kopylova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. I. Komech
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Kopylova.

Additional information

Communicated by P.-L. Lions

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kopylova, E., Komech, A.I. On Asymptotic Stability of Kink for Relativistic Ginzburg–Landau Equations. Arch Rational Mech Anal 202, 213–245 (2011). https://doi.org/10.1007/s00205-011-0415-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-011-0415-1

Keywords

Search

Navigation