Skip to main content
SpringerLink
Log in

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.

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. Albiez M., Gati R., Fölling J., Hunsmann S., Cristiani M., Oberthaler M.K.: Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction. Phys. Rev. Lett. 95, 010402 (2005)

    Article  ADS  Google Scholar 

  2. Aschbacher W.H., Fröhlich J., Graf G.M., Schnee K., Troyer M.: Symmetry breaking regime in the nonlinear hartree equation. J. Math. Phys. 43, 3879–3891 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Buffoni, B., Toland, J.: Analytic Theory of Global Bifurcation. Princeton, NJ: Princeton University Press, 2003

  4. Cambournac C., Sylvestre T., Maillotte H., Vanderlinden B., Kockaert P., Emplit Ph., Haelterman M.: Symmetry-breaking instability of multimode vector solitons. Phys. Rev. Lett. 89, 083901 (2002)

    Article  ADS  Google Scholar 

  5. Carretero-González R., Frantzeskakis D.J., Kevrekidis P.G.: Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques. Nonlinearity 21, R139–R202 (2008)

    Article  ADS  MATH  Google Scholar 

  6. Cazenave, T.: Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics Vol. 10. New York: Courant Institute, 2003

  7. Cuccagna S.: On asymptotic stability in energy space of ground states of NLS in 1D. J. Diff. Eqs. 245, 653–691 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69, 397–408 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grillakis M.: Linearized instability for nonlinear Schrödinger and Klein–Gordon equations. Comm. Pure Appl. Math. 41, 747–774 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grillakis M., Shatah J., Strauss W.: Stability theory of solitary waves in the presence of symmetry. J. Funct. Anal. 74, 160–197 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jackson R.K., Weinstein M.I.: Geometric analysis of bifurcation and symmetry breaking in a Gross-Pitaevskii equation. J. Stat. Phys. 116, 881–905 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Jeanjean H., Stuart C.: Nonlinear eigenvalue problems having an unbounded branch of symmetric bound states. Adv. Diff. Eqs. 4, 639–670 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Jeanjean H., Lucia M., Stuart C.: Branches of solutions to semilinear elliptic equations on \({\mathbb{R}^N}\) . Math. Z. 230, 79–105 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jeanjean H., Lucia M., Stuart C.: The branch of positive solutions to a semilinear elliptic equation on \({\mathbb{R}^N}\) . Rend. Sem. Mat. Univ. Padova 101, 229–262 (1999)

    MathSciNet  MATH  Google Scholar 

  15. Joannopoulos, J.D., Johnson, S.G., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton, NJ, Princeton University Press, 2008

  16. Kevrekidis P.G., Chen Z., Malomed B.A., Frantzeskakis D.J., Weinstein M.I.: Spontaneous symmetry breaking in photonic lattices : Theory and experiment. Phys. Lett. A 340, 275–280 (2005)

    Article  ADS  MATH  Google Scholar 

  17. Kirr E.W., Kevrekidis P.G., Shlizerman E., Weinstein M.I.: Symmetry-breaking bifurcation in nonlinear Schrödinger/Gross–Pitaevskii equations. SIAM J. Math. Anal. 40, 56–604 (2008)

    Article  MathSciNet  Google Scholar 

  18. Kirr E.W., Zarnescu A.: Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases. J. Diff. Eqs. 247, 710–735 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kirr E.W., Mızrak Ö.: Asymptotic stability of ground states in 3d nonlinear Schrödinger equation including subcritical cases. J. Funct. Anal. 257, 3691–3747 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kivshar, Yu.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. San Diego, CA: Academic Press, 2003

  21. Marzuola J.L., Weinstein M.I.: Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross–Pitaevskii equations. Discr. Cont. Dynam. Syst. A 28, 1505–1554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mizumachi T.: Asymptotic stability of small solitary waves to 1D nonlinear Schrödinger equations with potential. J. Math. Kyoto Univ. 48, 471–497 (2008)

    MathSciNet  MATH  Google Scholar 

  23. Nirenberg, L.: Topics in nonlinear functional analysis. Courant Lecture Notes 6, New York: Courant Inst, 2001

  24. Oh Y.-G.: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 131(2), 223–253 (1990)

    Article  ADS  MATH  Google Scholar 

  25. Ornigotti M., Della Valle G., Gatti D., Longhi S.: Topological suppression of optical tunneling in a twisted annular fiber. Phys. Rev. A 76, 023833 (2007)

    Article  ADS  Google Scholar 

  26. Pethick, C.J., Smith, H.: Bose-Einstein condensation in dilute gases. Cambridge: Cambridge University Press, 2002

  27. Pillet C.A., Wayne C.E.: Invariant manifolds for a class of dispersive, Hamiltonian, partial differential equations. J. Diff. Eqs. 141, 310–326 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pitaevskii L.P., Stringari S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  29. Rose H.A., Weinstein M.I.: On the bound states of the nonlinear Schrödinger equation with a linear potential. Physica D 30, 207–218 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. Sacchetti, A.: Universal critical power for nonlinear Schrödinger equations with a symmetric double well potential. Phys. Rev. Lett. 103, 194101 (4 pages) (2009)

    Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Weinstein M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39, 51–68 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana–Champaign, Urbana, IL, 61801, USA

    E. Kirr

  2. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    P. G. Kevrekidis

  3. Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    D. E. Pelinovsky

Authors
  1. E. Kirr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. G. Kevrekidis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. E. Pelinovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Kirr.

Additional information

Communicated by P. Constantin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kirr, E., Kevrekidis, P.G. & Pelinovsky, D.E. Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials. Commun. Math. Phys. 308, 795–844 (2011). https://doi.org/10.1007/s00220-011-1361-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1361-3

Keywords

Search

Navigation