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Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

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An Erratum to this article was published on 16 April 2013

Abstract

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed-point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one-dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.

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References

  • Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Grundlehren der mathematischen Wissenschaften—A series of Comprehensive Studies in Mathematics, vol. 251. Springer, New York (1982)

    Book  Google Scholar 

  • Christodoulides, D.N., Lederer, F., Silberberg, Y.: Discretizing light behavious in linear and nonlinear waveguide lattices. Nature 424, 817–823 (2003)

    Article  Google Scholar 

  • Claude, Ch., Kivshar, Yu.S., Kluth, O., Spatschek, K.H.: Moving localized modes in nonlinear lattices. Phys. Rev. B 47, 14228 (1993)

    Article  Google Scholar 

  • Cuevas, J., Eilbeck, J.C., Karachalios, N.I.: Thresholds for time periodic solutions on the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete Contin. Dyn. Syst. 21, 445–475 (2008a)

    Article  MathSciNet  MATH  Google Scholar 

  • Cuevas, J., Eilbeck, J.C., Karachalios, N.: A lower bound for the power of periodic solutions of the defocusing discrete nonlinear Schrödinger equation. Dyn. Partial Differ. Equ. 5, 69–85 (2008b)

    MathSciNet  MATH  Google Scholar 

  • Cuevas, J., Karachalios, N., Palmero, F.: Lower and upper estimates on the excitation threshold for breathers in discrete nonlinear Schrödinger lattices. J. Math. Phys. 50, 112705 (2009)

    Article  MathSciNet  Google Scholar 

  • Cuevas, J., Karachalios, N., Palmero, F.: Energy thresholds for the existence of breather solutions and traveling waves on lattices. Appl. Anal. 89, 1351–1385 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Dorignac, J., Zhou, J., Campbell, D.K.: Discrete breathers in nonlinear Schrödinger hypercubic lattices with arbitrary power nonlinearity. Physica D 237, 486–504 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Eilbeck, J.C., Johansson, M.: In: Vázquez, L., MacKay, R.S., Zorzano, M.P. (eds.) Localization and Energy Transfer in Nonlinear Systems, pp. 44–67. World Scientific, Singapore (2003)

    Chapter  Google Scholar 

  • Falvo, C., Pouthier, V., Eilbeck, J.C.: Fast energy transfer mediated by multi-quanta bound states in a nonlinear quantum lattice. Physica D 221, 58–71 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Flach, S., Gorbach, A.V.: Discrete breathers—advances in theory and applications. Phys. Rep. 467, 1–116 (2008)

    Article  Google Scholar 

  • Flach, S., Willis, C.R.: Discrete breathers. Phys. Rep. 295, 181–264 (1998)

    Article  MathSciNet  Google Scholar 

  • Flach, S., Kladko, K., MacKay, R.: Energy thresholds for discrete breathers in one-, two-, and three-dimensional lattices. Phys. Rev. Lett. 78, 1207–1210 (1997)

    Article  Google Scholar 

  • Haskins, M., Speight, J.M.: Breather initial profiles in chains of weakly coupled anharmonic oscillators. Phys. Lett. A 299, 549–557 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Hennig, D., Tsironis, G.: Wave transmission in nonlinear lattices. Phys. Rep. 307, 333–432 (1999)

    Article  MathSciNet  Google Scholar 

  • Herrmann, M.: Homoclinic standing waves in focusing DNLS equations. Discrete Contin. Dyn. Syst. 31, 737–752 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Johansson, M.: Discrete nonlinear Schrödinger approximation of a mixed Klein–Gordon/Fermi–Pasta–Ulam chain: modulational instability and a statistical condition for creation of thermodynamic breathers. Physica D 216, 62–70 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Karachalios, N.I.: A remark on the existence of breather solutions for the discrete nonlinear Schrödinger equation: the case of site dependent anharmonic parameter. Proc. Edinb. Math. Soc. 49, 115–129 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Kastner, M.: Energy thresholds for discrete breathers. Phys. Rev. Lett. 92(10), 104301 (2004)

    Article  Google Scholar 

  • Kevrekidis, P.G. (ed.): The Discrete Nonlinear Schrödinger Equation. Mathematical Analysis, Numerical Computations and Physical Perspectives. Springer, Berlin (2009)

    MATH  Google Scholar 

  • Kevrekidis, P.G., Frantzeskakis, D.J.: Pattern forming dynamical instabilities of Bose–Einstein condensates. Mod. Phys. Lett. B 18, 173–202 (2004)

    Article  Google Scholar 

  • Kevrekidis, P.G., Rasmussen, K.Ø., Bishop, A.R.: The discrete nonlinear Schrödinger equation: a survey of recent results. Int. J. Mod. Phys. B 15, 2833–2900 (2001)

    Article  Google Scholar 

  • Kevrekidis, P.G., Susanto, H., Chen, Z.: High-order-mode soliton structures in two-dimensional lattices with defocusing nonlinearity. Phys. Rev. E 74, 066606 (2006)

    Article  MathSciNet  Google Scholar 

  • Kevrekidis, P.G., Frantzeskakis, D.J., Carretero-González, R. (eds.): Emergent Nonlinear Phenomena in Bose–Einstein Condensates: Theory and Experiment. Springer Series on Atomic, Optical, and Plasma Physics, vol. 45 (2008)

    MATH  Google Scholar 

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

    Google Scholar 

  • Konotop, V.V., Brazhnyi, V.A.: Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B 18, 627–651 (2004)

    Article  Google Scholar 

  • Kundu, K.: Perturbative study of classical Ablowitz–Ladik type soliton dynamics in relation to energy transport in a-helical proteins. Phys. Rev. E 61, 5839 (2000)

    Article  MathSciNet  Google Scholar 

  • Lederer, F., Stegeman, G.I., Christodoulides, D.N., Assanto, G., Segev, M., Silberberg, Y.: Discrete solitons in optics. Phys. Rep. 463, 1–126 (2008)

    Article  Google Scholar 

  • Morsch, O., Oberthaler, M.: Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179–210 (2006)

    Article  Google Scholar 

  • Öster, M., Johansson, M.: Phase twisted modes and current reversals in a lattice model of waveguide arrays with nonlinear coupling. Phys. Rev. E 71, 025601(R) (2005)

    Article  Google Scholar 

  • Öster, M., Johansson, M., Eriksson, A.: Enhanced mobility of strongly localized modes in waveguide arrays by inversion of stability. Phys. Rev. E 67, 056606 (2003)

    Article  Google Scholar 

  • Öster, M., Gaididei, Y.B., Johansson, M., Christiansen, P.: Nonlocal and nonlinear dispersion in a nonlinear Schrödinger-type equation: exotic solitons and short-wavelength instabilities. Physica D 198, 29–50 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Pelinovsky, D.E.: Translationally invariant nonlinear Schrödinger lattices. Nonlinearity 19, 2695–2716 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Penati, T., Paleari, S.: Breathers and Q-Breathers: two sides of the same coin. SIAM J. Appl. Dyn. Syst. 11, 1–30 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Peyrard, M.: Nonlinear dynamics and statistical physics of DNA. Nonlinearity 17, R1–R40 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Sato, M., Hubbard, B.E., Sievers, A.J.: Nonlinear energy localization and its manipulation in micromechanical oscillator arrays. Rev. Mod. Phys. 78, 137–157 (2006)

    Article  Google Scholar 

  • Weinstein, M.: Excitation thresholds for nonlinear localized modes on lattices. Nonlinearity 12, 673–691 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Zeidler, E.: Linear Monotone Operators. Nonlinear Functional Analysis and Its Applications, vol. II/A. Springer, New York (1990)

    MATH  Google Scholar 

  • Zeidler, E.: Applied Functional Analysis: Applications to Mathematical Physics. Applied Mathematical Sciences, vol. 108. Springer, New York (1995)

    Google Scholar 

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Acknowledgements

We are indebted to Chris Eilbeck for having suggested us to consider the model presented in the manuscript. B.S.R. and J.C. acknowledge financial support from the MICINN project FIS2008-04848. P.G.K. gratefully acknowledges support from the US National Science Foundation via grants NSF-DMS-0806762 and NSFCMMI-1000337, from the US Air Force via award FA9550-12-1-0332, as well as from the Alexander von Humboldt Foundation, the Alexander S. Onassis Public Benefit Foundation via grant RZG 003/2010-2011 and the Binational Science Foundation via grant 2010239.

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Authors and Affiliations

  1. Department of Mathematics, University of the Aegean, Karlovassi, 83200, Samos, Greece

    N. I. Karachalios

  2. Grupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla, C/Virgen de Africa, 7, 41011, Sevilla, Spain

    B. Sánchez-Rey & J. Cuevas

  3. Department of Mathematics and Statistics, University of Massachusetts, Lederle Graduate Research Tower, Amherst, MA, 01003-9305, USA

    P. G. Kevrekidis

Authors
  1. N. I. Karachalios
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  2. B. Sánchez-Rey
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  3. P. G. Kevrekidis
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  4. J. Cuevas
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Correspondence to N. I. Karachalios.

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Communicated by P. Newton.

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Karachalios, N.I., Sánchez-Rey, B., Kevrekidis, P.G. et al. Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping. J Nonlinear Sci 23, 205–239 (2013). https://doi.org/10.1007/s00332-012-9149-y

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  • DOI: https://doi.org/10.1007/s00332-012-9149-y

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