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Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations pp 639–655Cite as

Conditions and Stability Analysis for Saddle-Node Bifurcations of Solitary Waves in Generalized Nonlinear Schrödinger Equations

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Part of the book series: Progress in Optical Science and Photonics ((POSP,volume 1))

Abstract

Saddle-node bifurcations of solitary waves in generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions are analyzed. First, general conditions for these bifurcations are derived. Second, it is shown analytically that the linear stability of these solitary waves does not switch at saddle-node bifurcations, which is in stark contrast with finite-dimensional dynamical systems where stability switching takes place. Third, it is shown that this absence of stability switching does not contradict the Vakhitov–Kolokolov stability criterion or the results in finite-dimensional dynamical systems. Fourth, it is shown that this absence of stability switching holds not only for real potentials but also for complex potentials. Lastly, various numerical examples will be given to confirm these analytical findings. In particular, saddle-node bifurcations with both branches of solitary waves being stable will be presented.

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References

  1. J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. (Springer, New York, 1990)

    Google Scholar 

  2. B. Buffoni, A.R. Champneys, J.F. Toland, Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system, J. Dyn. Differ. Equ. 8, 221 (1996)

    Google Scholar 

  3. T.S. Yang, T.R. Akylas, On asymmetric gravitycapillary solitary waves. J. Fluid Mech. 30, 215 (1997)

    Google Scholar 

  4. M. Chen, Solitary-wave and multi-pulsed travelling-wave solutions of Boussinesq systems. Appl. Anal. 75, 213 (2000)

    Google Scholar 

  5. J. Burke, E. Knobloch, Homoclinic snaking: Structure and stability. Chaos 17, 037102 (2007)

    Google Scholar 

  6. G. Herring, P.G, Kevrekidis, R. Carretero–Gonzalez, B.A. Malomed, D.J. Frantzeskakis, A.R. Bishop. Trapped bright matter–wave solitons in the presence of localized inhomogeneities, Phys. Lett. A 345, 144 (2005)

    Google Scholar 

  7. T. Kapitula, P. Kevrekidis, Z. Chen, Three is a crowd: Solitary waves in photorefractive media with three potential wells. SIAM J. Appl. Dyn. 5, 598 (2006)

    Google Scholar 

  8. A. Sacchetti, Universal critical power for nonlinear Schrodinger equations with symmetric double well potential. Phys. Rev. Lett. 103, 194101 (2009)

    Google Scholar 

  9. T.R. Akylas, G. Hwang, J. Yang, From nonlocal gap solitary waves to bound states in periodic media. Proc. Roy. Soc. A 468, pp. 116–135 (2012)

    Google Scholar 

  10. I.M. Merhasin, B.V. Gisin, R. Driben, B.A. Malomed, Finite-band solitons in the Kronig–Penney model with the cubic-quintic nonlinearity. Phys. Rev. E 71, 016613 (2005)

    Google Scholar 

  11. F. Dalfovo, S. Giorgini, L.P. Pitaevskii, S. Stringari, Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463 (1999)

    Google Scholar 

  12. Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)

    Google Scholar 

  13. J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems (SIAM, Philadelphia, 2010)

    Google Scholar 

  14. Y.V. Kartashov, B.A. Malomed, L. Torner, Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247–306 (2011)

    Google Scholar 

  15. N.G. Vakhitov, A.A. Kolokolov, Stationary solutions of the wave equation in a medium with nonlinearity saturation, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 16, p. 1020. [Radiophys. Quantum Electron. 16, p. 783 (1973)]

    Google Scholar 

  16. Z. Birnbaum, B.A. Malomed, Families of spatial solitons in a two-channel waveguide with the cubic–quintic nonlinearity. Physica D 237, 3252 (2008)

    Google Scholar 

  17. C.M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998)

    Google Scholar 

  18. S. Nixon, L. Ge, J. Yang, Stability analysis for solitons in PT-symmetric optical lattices, Phys. Rev. A85, 023822 (2012)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, 05401, USA

    Jianke Yang

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  1. Jianke Yang
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Correspondence to Jianke Yang .

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Editors and Affiliations

  1. Faculty of Engineering The Iby and Aladar Fleischman, Tel Aviv University, Ramat Aviv, Israel

    Boris A. Malomed

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Yang, J. (2012). Conditions and Stability Analysis for Saddle-Node Bifurcations of Solitary Waves in Generalized Nonlinear Schrödinger Equations. In: Malomed, B. (eds) Spontaneous Symmetry Breaking, Self-Trapping, and Josephson Oscillations. Progress in Optical Science and Photonics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10091_2012_3

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  • DOI: https://doi.org/10.1007/10091_2012_3

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21206-2

  • Online ISBN: 978-3-642-21207-9

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