We have simulated numerically the behavior of the one-dimensional, periodic FPU-alpha and Toda lattices to optical and acoustic initial excitations of small-but finite and large amplitudes. For the small-through-intermediate amplitudes (small initial energy per particle) we find nearly recurrent solutions, where the acoustic result is due to the appearance of solitons and where the optical result is due to the appearance of localized breather-like packets. For large amplitudes, we find complex-but-regular behavior for the Toda lattice and “stochastic” or chaotic behaviors for the alpha lattice. We have used the well-known diagnostics: Localization parameter; Lyapounov exponent, and slope of a linear fit to linear normal mode energy spectra. Space-time diagrams of local particle energy and a wave-related quantity, a discretized Riemann invariant are also shown. The discretized Riemann invariants of the alpha lattice reveal soliton and near-soliton properties for acoustic excitations. Except for the localization parameter, there is a clear separation in behaviors at long-time between integrable and nonintegrable systems.
REFERENCES
1.
Berman
, G. P.
, and Izrailev
, F. M.
(2005
). “The Fermi-Pasta-Ulam problem: Fifty years of progress
,” Chaos
15
, 015104
.2.
Budinsky
, N.
, and Bountis
, T.
(1983
). “Stability of nonlinear modes and chaotic properties of 1D Fermi-Pasta-Ulam lattices
,” Physica D
8
, 445
–452
.3.
Casetti
, L.
, Cerruti-Sola
, M.
, Pettini
, M.
, and Cohen
, E. G. D.
(1997
). “The Fermi-Pasta-Ulam problem revisited: Stochasticity thresholds in nonlinear Hamiltonian systems
,” Phys. Rev. E
55
, 6566
–6574
.4.
Chechin
, G. M.
, Novikova
, N. V.
, and Abramenko
, A. A.
(2002
). “Bushes of vibrational modes for Fermi-Pasta-Ulam chains
,” Physica D
166
, 208
–238
.5.
Chechin
, G. M.
, Ryabov
, D. S.
, and Zhukov
, K. G.
(2004
). “Stability of low-dimensional bushes of vibrational modes in the Fermi-Pasta-Ulam chains
”, http://arxiv.org/PS_cache/nlin/pdf/0403/0403040.pdf6.
Dauxois
, T.
, Khomeriki
, R.
, Piazza
, F.
, and Ruffo
, S.
(2005
). “The anti-FPU problem
,” Chaos
15
, 015110
.7.
Dauxois
, T.
, Ruffo
, S.
, and Torcini
, A.
(1997
). “Modulational estimate for the maximal Lyapunov exponent in Fermi-Pasta-Ulam chains
,” Phys. Rev. E
56
, R6229
–R6232
.8.
Dauxois
, T.
, Ruffo
, S.
, and Torcini
, A.
(1998
). “Analytical estimation of the maximal Lyapunov exponent in oscillator chains
,” J. Phys. IV
8
, Pr. 6, 147
–156
.9.
Deem
, G. S.
, Zabusky
, N. J.
, and Kruskal
, M. D.
(1965
). Formation, propagation, and interaction of solitons in nonlinear dispersive media. This film had three parts: (1) Solution of the KdV equation with a sinusoidal initial condition; (2) Solution of the modified-KdV equation with a compressive and rarefactive soliton initial condition; (3) Solution of the nonlinear lattice with a high-frequency (“optical”) initial condition. A film that was available on free loan from the Bell Telephone Labs film library for over and will soon be available on the internet at my URL.10.
Fermi
,E.
, Pasta
,J.
, and Ulam
,S. M.
(1955
). “Studies of nonlinear problems
,” LANL Report, No. 1940;see Fermi’s collected works or Stanislaw Ulam
Sets, Numbers, and Universes
, Selected Papers (MIT Press
, Cambridge
, 1974
).11.
Izrailev
, F. M.
, and Chirikov
, B. V.
(1966
). “Statistical properties of a nonlinear string
,” in Russian, Sov. Phys. Dokl.
11
, 30
(1965).12.
Kruskal
, M. D.
, and Zabusky
, N. J.
(1964
). “Stroboscopic perturbation procedure for treating a class of nonlinear wave equations
,” J. Math. Phys.
2
, 231
–244
.13.
Pettini
, M.
, Casetti
, L.
, Cerruti-Sola
, M.
, Franzosi
, R.
, and Cohen
, E. G. D.
(2005
). “Weak and strong chaos in Fermi-Pasta-Ulam models and beyond
,” Chaos
15
, 015106
.14.
Rink
,B.
(2001
). “Symmetry and resonance in periodic FPU chains
,” Commun. Math. Phys.
218
, 665
–685
;see also www.ma.ic.ac.uk/~brink
15.
Rink
, B.
(private communication, 2005).16.
Shelepansky
, D.
(1997
). “Low energy chaos in the Fermi-Pasta-Ulam problem
,” Nonlinearity
10
, 1331
–38
.17.
Toda
, M.
(1967a
). “Vibration of a Chain with Nonlinear Interaction
,” J. Phys. Soc. Jpn.
22
, 431
–436
.18.
19.
Toda
,M.
(1969
). “Mechanics and Statistical Mechanics of Nonlinear Chains
,” J. Phys. Soc. Jpn.
26
, Suppl. 235
–237
;Invited paper presented by M. Toda at the International Conference on Statistical Mechanics
, Kyoto, Japan, September, 1968.20.
Toda
, M.
(1989
). Theory of Nonlinear lattices
, 2nd ed. (Springer
, Berlin
).21.
Tuck
, J. L.
, and Menzel
, M. T.
(1972
). “The superperiod of the nonlinear weighted string (FPU) problem
,” Adv. Math.
9
, 399
.22.
Yoshida
, H.
(1990
). “Construction of higher order symplectic integrators
,” Phys. Lett. A
150
, no. 5–7
, 262
–268
.23.
Zabusky
, N. J.
, and Kruskal
, M. D.
(1965
). “Interaction of solitons in a collisionless plasma and the recurrence of initial states
,” Phys. Rev. Lett.
15
, 240
–243
.24.
Zabusky
, N. J.
(1962
). “Exact solution for the vibrations of a nonlinear continuous model string
,” J. Math. Phys.
5
, 497
–499
.25.
Zabusky
, N. J.
(1967
). “A synergetic approach to problems of nonlinear dispersive wave propagation and interaction
,” in Proceedings of the Symposium Nonlinear Partial Differential Equations
, edited by W.
Ames
(University of Delaware, Newark, Delaware, published by Academic Press
, New York, 1967). This invited paper was presented in 1965.26.
Zabusky
,N. J.
(1969
). “Nonlinear lattice dynamics and energy sharing
,” J. Phys. Soc. Jpn.
26
, Suppl., p. 196
;Invited paper presented at the International Conference on Statistical Mechanics
, Kyoto, Japan, September, 1968.27.
Zabusky
, N. J.
(1981
). “Computational synergetics and mathematical innovation
,” J. Comput. Phys.
43
, 195
–249
.28.
Zabusky
, N. J.
(2005
). “Fermi-Pasta-Ulam, solitons and the fabric of nonlinear and computational science: History, synergetics and visiometrics
,” Chaos
15
, 015102
.29.
Zabusky
, N. J.
, and Deem
, G. S.
(1967
). “Dynamics of nonlinear lattices I, localized optical excitations, acoustic radiation and strong nonlinear behavior
,” J. Comput. Phys.
2
, 126
–153
.© 2006 American Institute of Physics.
2006
American Institute of Physics
You do not currently have access to this content.
