Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

A. Cafarella; M. Leo; R. A. Leo

doi:10.1103/PhysRevE.69.046604

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Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

A. Cafarella, M. Leo, and R. A. Leo

Dipartimento di Fisica dell’Università, 73100 Lecce, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, Italy

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Phys. Rev. E 69, 046604 – Published 14 April, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.046604

Abstract

The stability of the one-mode nonlinear solutions of the Fermi-Pasta-Ulam $β$ system is numerically investigated. No external perturbation is considered for the one-mode exact analytical solutions, the only perturbation being that introduced by computational errors in the numerical integration of motion equations. The threshold energy for the excitation of the other normal modes and the dynamics of this excitation are studied as a function of the parameter $μ$ characterizing the nonlinearity, the energy density $ε$ and the number N of particles of the system. The results achieved confirm in part previous ones, obtained with a linear analysis of the problem of the stability, and clarify the dynamics by which a one-mode exchanges energy with the other modes with increasing energy density. In a range of energy density near the threshold value and for various values of the number of particles N, the nonlinear one-mode exchanges energy with the other linear modes for a very short time, immediately recovering all its initial energy. This sort of recurrence is very similar to Fermi recurrences, even if in the Fermi recurrences the energy of the initially excited mode changes continuously and only periodically recovers its initial value. A tentative explanation for this intermittent behavior, in terms of Floquet’s theorem, is proposed. Preliminary results are also presented for the Fermi-Pasta-Ulam $α$ system which show that there is a stability threshold, for large N, independent of N.

References (24)

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[c1] E. Fermi, J.R. Pasta, and S. Ulam, Collected Papers of E. Fermi, edited by E. Segre (University of Chicago Press, Chicago, 1965).

[c2] J. Ford, J. Math. Phys. 2, 387 (1961); J. Ford and J. Waters, 4, 1293 (1963).

[c3] E.A. Jackson, J. Math. Phys. 4, 551 (1963).

[c4] R.S. Northcote and R.B. Potts, J. Math. Phys. 5, 383 (1964).

[c5] F.M. Israiliev and B.V. Chirikov, Dokl. Akad. Nauk SSSR 166, 57 (1966) [Sov. Phys. Dokl. 11, 30 (1966)].

[c6] G.M. Zaslavsky and R.Z. Sagdeev, Zh. Éksp. Teor. Fiz. 52, 1081 (1967) [Sov. Phys. JETP 25, 718 (1967)].

[c7] N. J. Zabusky, Nonlinear Partial Differential Equations (Academic Press, New York, 1967), p. 223.

[c8] H. Poincaré, Les Methodes Nouvelles de la Méchanique Celeste (Blanchard, Paris, 1987), Vol. 3, p. 389.

[c9] E. Fermi, Nuovo Cimento 25, 267 (1923); 26, 105 (1923).

[c10] J. Ford, Phys. Rep. 213, 271 (1992).

[c11] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR 98, 527 (1954) [Lect. Notes Phys. 93, 51 (1979); V.I. Arnold, Russ. Math. Surveys 18, 9 (1963); J. Moser, Nachr. Adad. Wiss. Goett. II, Math.-Phys. K1.2 1, 1 (1962).

[c12] N. Saito, N. Ooyama, Y. Aizawa, and H. Hirooka, Suppl. Prog. Theor. Phys. 45, 209 (1970).

[c13] R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani, Phys. Rev. A 31, 1039 (1985).

[c14] M. Pettini and M. Landolfi, Phys. Rev. A 41, 768 (1990).

[c15] L. Casetti, M. Cerruti-Sola, M. Pettini, and E.G.D. Cohen, Phys. Rev. E 55, 6566 (1997).

[c16] H. Kantz, Physica D 39, 322 (1989).

[c17] R. Livi, M. Pettini, S. Ruffo, M. Sparpaglione, and A. Vulpiani, Phys. Rev. A 28, 3544 (1983).

[c18] P. Poggi and S. Ruffo, Physica D 103, 251 (1997).

[c19] N. Budinsky and T. Bountis, Physica D 8, 445 (1983).

[c20] S. Flach, Physica D 91, 223 (1996).

[c21] K.W. Sandusky and J.B. Page, Phys. Rev. B 50, 866 (1994).

[c22] L. Casetti, Phys. Scr. 51, 29 (1995).

[c23] L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon, London, 1960).

[c24] N. Fameli, F.L. Curzon, and S. Mikoshiba, Am. J. Phys. 67, 127 (1999).

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Numerical analysis of the one-mode solutions in the Fermi-Pasta-Ulam system

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