×

The discrete self-trapping equation. (English) Zbl 0583.34026

A simple system of ordinary differential equations is introduced which has an application to the dynamics of small molecules, molecular crystals, self-trapping in amorphous semiconductors, and globular proteins. Analytical, numerical and perturbation methods are used to study the properties of stationary solutions. General solution trajectories can be either sinusoidal, periodic, quasiperiodic or chaotic.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92B05 General biology and biomathematics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Landau, L. D., Phys. Zeit. Sowjetunion, 3, 644 (1933)
[2] Pekar, S., J. Phys. U.S.S.R., 10, 347 (1946)
[3] Frölich, H., Adv. in Phys., 3, 325 (1954)
[4] Holstein, T., Ann. Phys., 8, 343 (1959)
[5] De Raedt, H.; Lagendijk, A., Phys. Rev., B80, 1671 (1984)
[6] Sov. Phys. JETP, 34, 62 (1972)
[7] Hasegawa, A.; Kodama, Y., Proc. IEEE, 69, 1145 (1981)
[8] Larraza, A.; Putterman, S., Theory of nonpropagating hydrodynamic solitons, Phys. Lett. A, 103, 15 (1984) · Zbl 0626.76026
[9] Sov. Phys. Usp., 25, 899 (1982), and references therein
[10] MacNeil, L.; Scott, A. C., Physica Scripta, 29, 284 (1984)
[11] Careri, G.; Buontempo, U.; Galluzzi, F.; Scott, A. C.; Gratton, E.; Shyamsunder, E., Spectroscopic evidence for Davydov-like solitons in acetanilide, Phys. Rev., B30, 4689 (1984)
[12] Volkenstein, M. V., J. Theor. Biol., 34, 193 (1972)
[13] Green, D. E.; Ji, S., Proc. Nat. Acad. Sci. (USA), 69, 726 (1972)
[14] Lomdahl, P. S., Nonlinear dynamics of globular protein, (Adey, W. R.; Lawrence, A. F., Los Alamos Lab. Report LA-UR-83-2252. Los Alamos Lab. Report LA-UR-83-2252, Nonlinear Electrodynamics in Biological Systems (1984), Plenum: Plenum New York), 143
[15] Eilbeck, J. C.; Lomdahl, P. S.; Scott, A. C., Soliton structure in crystalline acetanilide, Phys. Rev., B30, 4703 (1984)
[16] Thouless, D. J., Phys. Rep., 13, 95 (1974)
[17] Allen, J. P.; Colvin, J. T.; Stinson, D. G.; Flynn, C. P.; Stapleton, H. J., Biophys. J., 38, 299 (1982)
[18] Alexander, S.; Laermans, C.; Orbach, R.; Rosenberg, H. M., Phys. Rev., B28, 4615 (1983)
[19] Scott, A. C.; Lomdahl, P. S.; Eilbeck, J. C., Between the local mode and normal mode limits, Chem. Phys. Lett., 113, 29 (1985) · Zbl 0583.34026
[20] Ellis, J. W., Trans. Faraday Soc., 25, 888 (1929)
[21] Sage, M. L.; Jortner, J., Adv. Chem. Phys., 47, 293 (1981)
[22] Collins, M. A., Adv. Chem. Phys., 53, 225 (1983)
[23] Kubicek, M., A.C.M. Trans. Math. Soft., 2, 98 (1976)
[24] J. Carr and J.C. Eilbeck, Stability of stationary solutions of discrete self-trapping equation (to appear in Phys. Lett.).; J. Carr and J.C. Eilbeck, Stability of stationary solutions of discrete self-trapping equation (to appear in Phys. Lett.).
[25] Dodd, R. K.; Eilbeck, J. C.; Gibbon, J. D.; Morris, H. C., Solitons and Nonlinear Wave Equations (1982), Academic Press: Academic Press London · Zbl 0496.35001
[26] Landau, L. D., Dok. Akad. Nauk. SSSR. Dok. Akad. Nauk. SSSR, C.R. Acad. Sci. USSR, 44, 311 (1944)
[27] Genesis 7: 11 through 8: 5.; Genesis 7: 11 through 8: 5. · Zbl 0462.60094
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.