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Introduction to the Discrete Self-Trapping Equation

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Davydov’s Soliton Revisited

Part of the book series: NATO ASI 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.

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References

  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.

    Article  ADS  Google Scholar 

  2. L Bernstein. Local modes and degenerate perturbation theory. 1989. (these proceedings).

    Google Scholar 

  3. D W Brown. When is a soliton? 1989. (these proceedings).

    Google Scholar 

  4. D W Brown and Z Ivic. Unification of polaron and soliton theories of exciton transport. 1989. (San Diego preprint).

    Google Scholar 

  5. J Carr and JC Eilbeck. Stability of stationary solutions of the discrete self-trapping equation. Phys. Letts. A, 109:201–204, 1985.

    Article  MathSciNet  ADS  Google Scholar 

  6. D N Christodoulides and RI Joseph. Discrete self-focusing in nonlinear arrays of coupled waveguides. Optics Letters, 13:794–796, 1988.

    Article  ADS  Google Scholar 

  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.

    Article  MathSciNet  ADS  Google Scholar 

  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. J C Eilbeck. Nonlinear vibrational modes in a hexagonal molecule, pages 41-51. Volume 12, Ukrainian Academy of Sciences, Kiev, 1987.

    Google Scholar 

  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. J C Eilbeck, PS Lomdahl, and AC Scott. The discrete self-trapping equation. Physica D: Nonlinear Phenomena, 16:318–338, 1985.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. J C Eilbeck, PS Lomdahl, and AC Scott. Soliton structure in crystalline acetanilide. Phys. Rev. B, 30:4703–4712, 1984.

    Article  ADS  Google Scholar 

  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. H Feddersen. Quantum and classical descriptions of chaos in the DST equation. 1989. (these proceedings).

    Google Scholar 

  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.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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.

    Article  ADS  MATH  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  18. V I Inozemtsev and NA Rostov. New integrable systems of interacting nonlinear waves. 1988. Dubna preprint E5-88-622.

    Google Scholar 

  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.

    Article  ADS  Google Scholar 

  20. V M Kenkre. The discrete nonlinear Schrödinger equation: nonadiabatic effects, finite temperature consequences, and experimental manifestations. 1989. (these proceedings).

    Google Scholar 

  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. 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.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  25. V A Kuprievich. On autolocalization of the stationary states in a finite molecular chain. Physica D, 14:395–402, 1985.

    Article  MathSciNet  ADS  Google Scholar 

  26. K Lindenberg. Vibron solitons. 1989. (these proceedings).

    Google Scholar 

  27. G J Morrison. Homoclinic chaos in the DST equation. 1989. (in preparation).

    Google Scholar 

  28. I Nussbaum. Non-steady solutions of the discrete self-trapping equation. Phys. Lett. A, 118:127–130, 1986.

    Article  ADS  Google Scholar 

  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.

    Article  ADS  Google Scholar 

  30. A C Scott. A nonresonant discrete self-trapping system. Physica Scripta. (in press).

    Google Scholar 

  31. A C Scott and JC Eilbeck. On the CH stretch overtones of benzene. Chem. Phys. Lett, 132:23–28, 1986.

    Article  ADS  Google Scholar 

  32. A C Scott, PS Lomdahl, and JC Eilbeck. Between the local mode and normal mode limits. Chem. Phys. Letts., 113:21–36, 1985.

    Article  ADS  Google Scholar 

  33. A C Scott and L MacNeil. Binding energy versus nonlinearity for a “small” stationary soliton. Phys. Lett. A., 98:87–88, 1983.

    Article  ADS  Google Scholar 

  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. G P Tsironis and VM Kenkre. Initial condition effects in the evolution of a nonlinear dimer. Phys. Lett. A., 127:209–212, 1988.

    Article  ADS  Google Scholar 

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© 1990 Springer Science+Business Media New York

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Eilbeck, J.C. (1990). Introduction to the Discrete Self-Trapping Equation. In: Christiansen, P.L., Scott, A.C. (eds) Davydov’s Soliton Revisited. NATO ASI Series, vol 243. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9948-4_38

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  • DOI: https://doi.org/10.1007/978-1-4757-9948-4_38

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-9950-7

  • Online ISBN: 978-1-4757-9948-4

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