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Introduction to the Discrete Self-Trapping Equation

  • J. C. Eilbeck
Part of the NATO ASI Series book series (NSSB, volume 243)

Abstract

The discrete self-trapping (DST) equation models a coupled system of classical or quantum anharmonic oscillators. In this paper we review the physical motivations for this model, and describe some of the known solutions of the equation. The aim of this paper is to provide a basic introduction to other contributions to this volume covering recent results on the DST equation and its applications.

Keywords

Stationary Solution Jacobian Elliptic Function Couple Waveguide Single Soliton Dime Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    P. Banacky and A Zajac. Theory of particle transfer dynamics in solvated molecular complexes: analytic solutions of the discrete time-dependent nonlinear Schrödinger equation. I. conservative system. Chem Phys., 123:267–276, 1988.ADSCrossRefGoogle Scholar
  2. [2]
    L Bernstein. Local modes and degenerate perturbation theory. 1989. (these proceedings).Google Scholar
  3. [3]
    D W Brown. When is a soliton? 1989. (these proceedings).Google Scholar
  4. [4]
    D W Brown and Z Ivic. Unification of polaron and soliton theories of exciton transport. 1989. (San Diego preprint).Google Scholar
  5. [5]
    J Carr and JC Eilbeck. Stability of stationary solutions of the discrete self-trapping equation. Phys. Letts. A, 109:201–204, 1985.MathSciNetADSCrossRefGoogle Scholar
  6. [6]
    D N Christodoulides and RI Joseph. Discrete self-focusing in nonlinear arrays of coupled waveguides. Optics Letters, 13:794–796, 1988.ADSCrossRefGoogle Scholar
  7. [7]
    L Cruzeiro-Hansson, PL Christiansen, and JN Elgin. Comment on “Self-trapping on a dimer: time-dependent solutions of a discrete nonlinear Schrödinger equation”. Phys. Rev. B., 37:7896–7897, 1988.MathSciNetADSCrossRefGoogle Scholar
  8. [8]
    L Cruzeiro-Hansson, H Feddersen, R Flesch, PL Christiansen, M Salerno, and AC Scott. Classical and quantum analysis of chaos in the discrete self-trapping equation. 1989. (submitted for publication).Google Scholar
  9. [9]
    J C Eilbeck. Nonlinear vibrational modes in a hexagonal molecule, pages 41-51. Volume 12, Ukrainian Academy of Sciences, Kiev, 1987.Google Scholar
  10. [10]
    J C Eilbeck. Numerical simulations of the dynamics of polypeptide chains and proteins. In Chikao Kawabata and AR Bishop, editors, Computer Analysis for Life Science— Progress and Challenges in Biological and Synthetic Polymer Research, pages 12-21, Ohmsha, Tokyo, 1986.Google Scholar
  11. [11]
    J C Eilbeck, PS Lomdahl, and AC Scott. The discrete self-trapping equation. Physica D: Nonlinear Phenomena, 16:318–338, 1985.MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. [12]
    J C Eilbeck, PS Lomdahl, and AC Scott. Soliton structure in crystalline acetanilide. Phys. Rev. B, 30:4703–4712, 1984.ADSCrossRefGoogle Scholar
  13. [13]
    J C Eilbeck and AC Scott. Theory and applications of the discrete self-trapping equation. In PL Christiansen and RD Parmentier, editors, Structure, Coherence and Chaos in Dynamical Systems, pages 139-159, Manchester University Press, 1989.Google Scholar
  14. [14]
    H Feddersen. Quantum and classical descriptions of chaos in the DST equation. 1989. (these proceedings).Google Scholar
  15. [15]
    S De Filippo, M Fusco Girard, and M Salerno. Avoided crossing and nearest-neighbour level spacings for the quantum DST equation. Nonlinearity, 2:477–487, 1989.MathSciNetADSzbMATHCrossRefGoogle Scholar
  16. [16]
    S De Filippo, M Fusco Girard, and M Salerno. Lyapunov exponents for the n = 3 discrete self-trapping equation. Physica D, 26:411–414, 1987.ADSzbMATHCrossRefGoogle Scholar
  17. [17]
    S De Filippo, M Fusco Girard, and M Salerno. Numerical evidence of a sharp order window in a Hamiltonian system. Physica D, 29:421–426, 1988.ADSCrossRefGoogle Scholar
  18. [18]
    V I Inozemtsev and NA Rostov. New integrable systems of interacting nonlinear waves. 1988. Dubna preprint E5-88-622.Google Scholar
  19. [19]
    J H Jensen, PL Christiansen, JN Elgin, JD Gibbon, and O Skovgaard. Correlation exponents for trajectories in the low-dimensional discrete self-trapping equation. Phys. Lett A, 110:429–431, 1985.ADSCrossRefGoogle Scholar
  20. [20]
    V M Kenkre. The discrete nonlinear Schrödinger equation: nonadiabatic effects, finite temperature consequences, and experimental manifestations. 1989. (these proceedings).Google Scholar
  21. [21]
    V M Kenkre. The quantum nonlinear dimer and extensions. In S Pnevmatikos, T Boun-tis, and St. Pnevmatikos, editors, Singular behaviour and nonlinear dynamics, World Publishers, 1989.Google Scholar
  22. [22]
    V M Kenkre and DK Campbell. Self-trapping on a dimer: time-dependent solutions of a discrete nonlinear Schrödinger equation. Phys. Rev. B, 34:4959–4961, 1986.ADSCrossRefGoogle Scholar
  23. [23]
    V M Kenkre and H-L Wu. Interplay of quantum phases and nonlinearity in the nonadiabatic dimer. Phys. Lett. A, 135:120–124, 1989.ADSCrossRefGoogle Scholar
  24. [24]
    E W Knapp and SF Fischer. A unified theory of electron transfer and internal conversion based on solitary electronic states. J. Chem. Phys., 90:354–365, 1988.ADSCrossRefGoogle Scholar
  25. [25]
    V A Kuprievich. On autolocalization of the stationary states in a finite molecular chain. Physica D, 14:395–402, 1985.MathSciNetADSCrossRefGoogle Scholar
  26. [26]
    K Lindenberg. Vibron solitons. 1989. (these proceedings).Google Scholar
  27. [27]
    G J Morrison. Homoclinic chaos in the DST equation. 1989. (in preparation).Google Scholar
  28. [28]
    I Nussbaum. Non-steady solutions of the discrete self-trapping equation. Phys. Lett. A, 118:127–130, 1986.ADSCrossRefGoogle Scholar
  29. [29]
    I Nussbaum and SF Fischer. Analytic treatment of localized, stationary states of the discrete self-trapping equation. Phys. Lett. A, 115:268–270, 1986.ADSCrossRefGoogle Scholar
  30. [30]
    A C Scott. A nonresonant discrete self-trapping system. Physica Scripta. (in press).Google Scholar
  31. [31]
    A C Scott and JC Eilbeck. On the CH stretch overtones of benzene. Chem. Phys. Lett, 132:23–28, 1986.ADSCrossRefGoogle Scholar
  32. [32]
    A C Scott, PS Lomdahl, and JC Eilbeck. Between the local mode and normal mode limits. Chem. Phys. Letts., 113:21–36, 1985.ADSCrossRefGoogle Scholar
  33. [33]
    A C Scott and L MacNeil. Binding energy versus nonlinearity for a “small” stationary soliton. Phys. Lett. A., 98:87–88, 1983.ADSCrossRefGoogle Scholar
  34. [34]
    S Takeno. A classical and quantum-mechanical theory of vibron solitons and kinks in open systems and their implication of biological energy transfer. 1989. (these proceedings).Google Scholar
  35. [35]
    G P Tsironis and VM Kenkre. Initial condition effects in the evolution of a nonlinear dimer. Phys. Lett. A., 127:209–212, 1988.ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • J. C. Eilbeck
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityEdinburghScotland

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