Low-Dimensional Solutions in the Quartic Fermi–Pasta–Ulam System

Shinohara, Susumu

doi:10.1143/jpsj.71.1802

Cited by 25 publications

(51 citation statements)

References 9 publications

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“…Finally, we would like to comment on two papers by Shinohara [16], devoted to finding "type I subsets of modes" (bushes, in our terminology) in nonlinear chains with fixed endpoints. The author does not use any symmetry-related method and prefers to analyze the specific structure of the dynamical equations (practically, in the manner similar to that of Poggi and Ruffo in [14]).…”

Section: Some Historymentioning

confidence: 99%

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Stability of low-dimensional bushes of vibrational modes in the Fermi–Pasta–Ulam chains

Chechin

Ryabov

Zhukov

2005

Physica D: Nonlinear Phenomena

124

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Bushes of normal modes represent the exact excitations in the nonlinear physical systems with discrete symmetries [Physica D 117 (1998) 43]. The present paper is the continuation of our previous paper [Physica D 166 (2002) 208], where these dynamical objects of new type were discussed for the monoatomic nonlinear chains. Here, we develop a simple crystallographic method for finding bushes in nonlinear chains and investigate stability of one-dimensional and two-dimensional vibrational bushes for both FPU-α and FPU-β models, in particular, of those revealed recently in [Physica D 175 (2003) 31].

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Section: Some Historymentioning

confidence: 99%

“…We would also like to note that the stability problem for the bushes of modes (invariant manifolds or "modes subsets of I type") is not considered in [16]. On the other hand, this problem seems to be important for treating the induction phenomenon discussed in those papers.…”

Section: Some Historymentioning

confidence: 99%

Stability of low-dimensional bushes of vibrational modes in the Fermi–Pasta–Ulam chains

Chechin

Ryabov

Zhukov

2005

Physica D: Nonlinear Phenomena

124

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“…Although the expressions have already been obtained for the lattice with quartic anharmonicity (the so-called FPU lattice) in Ref. 8), they are shown here to be applicable for the lattice with a general potential. We note that, in Ref.…”

Section: §1 Introductionmentioning

confidence: 89%

Low-Dimensional Subsystems in Anharmonic Lattices

Shinohara

2003

Prog. Theor. Phys. Suppl.

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In an anharmonic lattice model, when one initially places energy only in a subset of normal modes, E, the energy would not necessarily diffuse to all the modes, but can be retained by a certain subset of modes, I(⊃ E), called as the type I subset. This phenomenon is due to both the selection rules for the mode interaction and the initial localization of energy. For such initial excitation, the dynamics can be fully described by a low-dimensional sub-Hamiltonian. Analyzing the mode-coupling coefficients, we obtain useful expressions for the type I subsets for a one-dimensional monatomic lattice with fixed ends. In addition, we explicitly derive the one-and two-mode sub-Hamiltonians. §1. IntroductionOne of the basic problems in the study of anharmonic lattices is to understand the energy redistribution process among the normal modes. Although the anharmonicity violates the dynamical independence of the normal modes, it does not necessarily lead to the energy equipartition among them, which was firstly revealed in the numerical simulation by Fermi, Pasta and Ulam. 1) Since this pioneering work, much effort has been devoted to study how the relaxation properties depend on the degree of anharmonicity, the system size, the initial excitation of modes, and so on. 2) In numerical studies of the energy redistribution problem, the initial excitation is often given in the following form:where E k (0) is the energy initially placed in the k-th mode, and E is a subset of mode indices. is taken to be negligibly small but nonzero, which is unavoidable in numerical simulations. A well studied case is that of single mode excitation, 3)-6) in which the so-called "induction phenomenon" is observed; before violent energy exchange starts, the energy is mostly retained in the initially excited mode for a certain period. 3) With regard to the initial excitation (1 . 1), it is pointed out that in the → 0 limit, the energy would not necessarily spread over all the modes, but can be shared only among the modes in a certain subset I(⊃ E), named the type I subset. 7) From the phase space viewpoint, such initial excitation corresponds to placing an initial point upon a low-dimensional invariant submanifold. Therefore, the induction phenomenon, which occurs for nonzero , can be explained as the process * ) Present address:

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“…Note, that the stability of the π-mode (the bush B[â 2 ,î]) was discussed in a number of papers [18,19,20,8,9,10,12,13,14,15] by different methods and with an emphasis on different aspects of this stability. In particular, in our paper [8], a remarkable fact was revealed for the FPU-α chain: the stability threshold of the π-mode is one and the same for interactions with all the other modes of the chain.…”

Section: X(t) = {0 A(t) B(t) 0 −B(t) −A(t) | }mentioning

confidence: 99%

“…Let us note that stability of the π-mode in the FPU-α and FPU-β chains was investigated by different methods in a large number of papers (see, for example, [18,19,20,8,9,10,12,13,14,15]), but, to our best understanding, the influence of symmetry of these mechanical models on stability analysis was not discussed. Unlike the above cited works, the bush stability analysis presented in this paper based on the symmetry-related arguments only.…”

Section: Consideringmentioning

confidence: 99%

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

Chechin

Zhukov

2006

Phys. Rev. E

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We present a theorem that allows to simplify linear stability analysis of periodic and quasiperiodic nonlinear regimes in N -particle mechanical systems (both conservative and dissipative) with different kinds of discrete symmetry. This theorem suggests a decomposition of the linearized system arising in the standard stability analysis into a number of subsystems whose dimensions can be considerably less than that of the full system. As an example of such simplification, we discuss the stability of bushes of modes (invariant manifolds) for the Fermi-Pasta-Ulam chains and prove another theorem about the maximal dimension of the above mentioned subsystems.

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Low-Dimensional Solutions in the Quartic Fermi–Pasta–Ulam System

Cited by 25 publications

References 9 publications

Stability of low-dimensional bushes of vibrational modes in the Fermi–Pasta–Ulam chains

Stability of low-dimensional bushes of vibrational modes in the Fermi–Pasta–Ulam chains

Low-Dimensional Subsystems in Anharmonic Lattices

Stability analysis of dynamical regimes in nonlinear systems with discrete symmetries

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