Low-Dimensional Subsystems in Anharmonic Lattices

Shinohara, Susumu

doi:10.1143/ptps.150.423

Cited by 26 publications

(6 citation statements)

References 7 publications

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“…After the aforementioned paper by Poggi and Ruffo, several papers of other authors were published in which intermode interactions were also studied based on the specifics of interatomic interactions in the FPU chains [23][24][25][26]. Only in Refs.…”

Section: Theory Of the Bushes Of Nonlinear Normal Modesmentioning

confidence: 99%

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

Chechin¹,

Ryabov²

2021

Preprint

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Intermode interactions in one-dimensional nonlinear periodic structures have been studied by many authors, starting with the classical work by Fermi, Pasta, and Ulam (FPU) in the middle of the last century. However, symmetry selection rules for the energy transfer between nonlinear vibrational modes of different symmetry, which lead to the possibility of excitation of some bushes of such modes, were not revealed. Each bush determines an exact solution of nonlinear dynamical equations of the considered system. The collection of modes of a given bush does not change in time, while there is a continuous energy exchange between these modes. Bushes of nonlinear normal modes (NNMs) are constructed with the aid of group-theoretical methods and therefore they can exist for the case of large amplitude atomic vibrations and for any type of interatomic interactions. In most publications, bushes of NNMs or similar dynamical objects in one-dimensional systems are investigated under periodic boundary conditions. In this paper, we present a detailed study of the bushes of NNMs in monoatomic chains for the case of fixed boundary conditions, which sheds light on a series of new properties of the intermode interactions in such systems. We prove some theorems that justify a method for constructing bushes of NNMs by continuation of conventional normal modes to the case of large atomic oscillations. Our study was carried out for FPU chains, for the chains with the Lennard-Jones interatomic potential, as well as for the carbon chains (carbynes) in the framework of the density functional theory. For one-dimensional bushes (Rosenberg nonlinear normal modes), the amplitude-frequency diagrams are presented and the possibility of their modulational instability is briefly discussed. We also argue in favor of the fact that our methods and main results are valid for any monoatomic chain.

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Section: Theory Of the Bushes Of Nonlinear Normal Modesmentioning

confidence: 99%

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

Chechin¹,

Ryabov²

2021

Preprint

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“…A recent series of papers [3][4][5][6][7] has been devoted to the stability properties of an interesting class of solutions admitted by the FPU β system with periodic boundary conditions: the one-mode solutions (OMSs) [8][9][10]. These are exact solutions, usually referred to by means of the values of the mode number 3 4 N , where N is the number of particles of the system.…”

Section: Introductionmentioning

confidence: 99%

Non-Maxwellian behavior and quasistationary regimes near the modal solutions of the Fermi-Pasta-Ulamβsystem

Leo¹,

Leo²,

Tempesta³

et al. 2012

Phys. Rev. E

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In a recent paper [M. Leo, R. A. Leo, and P. Tempesta, J. Stat. Mech. (2011) P03003], it has been shown that the π/2-mode exact nonlinear solution of the Fermi-Pasta-Ulam β system, with periodic boundary conditions, admits two energy density thresholds. For values of the energy density ε below or above these thresholds, the solution is stable. Between them, the behavior of the solution is unstable, first recurrent and then chaotic. In this paper, we study the chaotic behavior between the two thresholds from a statistical point of view, by analyzing the distribution function of a dynamical variable that is zero when the solution is stable and fluctuates around zero when it is unstable. For mesoscopic systems clear numerical evidence emerges that near the second threshold, in a large range of the energy density, the numerical distribution is fitted accurately with a q-Gaussian distribution for very large integration times, suggesting the existence of a quasistationary state possessing a weakly chaotic behavior. A normal distribution is recovered in the thermodynamic limit.

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“…Hence, the following question naturally arises: Are there any other such exact NNMs in the FPU-β chain, under p.b.c.? Some examples of such modes are already known in the literature [Poggi & Ruffo, 1997;Rink, 2003;Shinohara, 2002Shinohara, , 2003Yoshimura, 2004;Leo & Leo , 2007;Antonopoulos & Bountis, 2006;Kosevich, 1993], under a terminology that differs among authors. Below we list these exact states in detail:…”

Section: Introductionmentioning

confidence: 99%

“…Many aspects of existence and stability of NNMs have been discussed in the literature (see, for example, [Chechin et al, 2002[Chechin et al, , 2005Rink, 2003;Poggi & Ruffo, 1997;Shinohara, 2002Shinohara, , 2003Yoshimura, 2004;Leo & Leo , 2007;Antonopoulos & Bountis, 2006;Budinsky & Bountis, 1983;Sandusky & Page, 1994;Kosevich, 1993]. What is important at this point is to pose certain fundamental questions concerning such NNMs, which we shall proceed to answer in this review to the best of our ability: Q1) Is the list ( 14) -( 19) of NNMs for the FPU-β chain complete?…”

Section: Introductionmentioning

confidence: 99%

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Discrete Symmetry and Stability in Hamiltonian Dynamics

Bountis,

Chechin,

Sakhnenko

2010

Preprint

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In the present tutorial we address a problem with a long history, which remains of great interest to date due to its many important applications: It concerns the existence and stability of periodic and quasiperiodic orbits in N -degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study is what we call nonlinear normal modes (NNMs), i.e periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine questions concerning the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions. In the periodic case, we find it particularly useful to approach the problem through the discrete symmetries of many models, employing group theoretical concepts to identify a special type of NNMs which we call one-dimensional "bushes". We then describe how to use linear combinations of s ≥ 2 such NNMs to construct s-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems including particle chains, a square molecule and octahedral crystals in 1,2 and 3 dimensions. Next, we exploit the symmetries of the linearized equations of motion about these bushes to demonstrate how they may be simplified to study the destabilization of these orbits, as a result of their interaction with NNMs not belonging to the same bush. Applying this theory to the famous Fermi Pasta Ulam (FPU) chain, we review a number of interesting results concerning the stability of NNMs and higher-dimensional bushes, which have appeared in the recent literature. We then turn to a newly developed approach to the analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions of our system. Using this approach, we demonstrate that the well-known "paradox" of FPU recurrences may in fact be explained in terms of the exponential localization of the energies E q of NNM's being excited at the low part of the frequency spectrum, i.e. q = 1, 2, 3, .... These results indicate that it is the stability of these low-dimensional compact manifolds called q-tori, that is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, expressed by a spectrum of indices called GALI k , 1 2 T. Bountis, G. Chechin and V. Sakhnenko k = 2, ..., 2N , by means of which one can determine accurately and efficiently the destabilization of q tori, leading, after very long times, to the breakdown of recurrences and, ultimately, to the equipartition of energy, at high enough values of the total energy E.

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Low-Dimensional Subsystems in Anharmonic Lattices

Cited by 26 publications

References 7 publications

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

Non-Maxwellian behavior and quasistationary regimes near the modal solutions of the Fermi-Pasta-Ulamβsystem

Discrete Symmetry and Stability in Hamiltonian Dynamics

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