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Stability of low-dimensional solutions in the Fermi-Pasta-Ulam system

Susumu Shinohara

Department of Physical Sciences
Ritsumeikan University, Japan

The Fermi-Pasta-Ulam (FPU) system represents a one-dimensional array of equi-mass particles coupled by nonlinear springs. Since the pioneering work of FPU [1], a large amount of numerical experiments have been carried out with this system in order to elucidate how nonlinearity leads to ergodicity and the equipartition law of energy. One of the striking phenomena revealed in these experiments is the "induction phenomenon" in the energy exchange process among normal modes, i.e., initially excited modes retain energy over a certain period before violent energy exchange process sets in [2]. The FPU system possesses low-dimensional solutions whose dynamics can be completely described by reduced Hamiltonians. Since the growth of initially non-excited modes can be attributed to the instability of the low-dimensional solutions, identification of them is important for precise understanding of the induction phenomenon. One can systematically construct the reduced Hamiltonians by introducing in the mode number space the "type I subsets" defined by Poggi and Ruffo [3]. In this presentation, we first present general expressions for the type I subsets, which, as far as we numerically checked, cover all of the possible type I subsets in the FPU system [4]. Using these expressions, we next construct the reduced Hamiltonians and investigate the stability of their solutions under the full system's dynamics, where the linear stability of these solutions is described by coupled Hill's equations. We report the independence of the stability properties on the phase space structure of the reduced Hamiltonian systems. Finally, we discuss the conditions for the occurrence of the induction phenomenon in Hamiltonian systems.

[1] E. Fermi, J. Pasta and S. Ulam, Los Alamos Report LA1940 (1955).
[2] N. Saito, N. Hirotomi and A. Ichimura, J. Phys. Soc. Jpn. 39 (1975) 1431.
[3] P. Poggi and S. Ruffo, Physica D103 (1997) 251.
[4] S. Shinohara, submitted to J. Phys. Soc. Jpn (2002).

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Next: Simon Up: Abstracts Previous: Selyshchev