Delocalized nonlinear vibrational modes of triangular lattices

Ryabov, Denis; Chechin, G. M.; Upadhyaya, Abhisek; Korznikova, Elena A.; Dubinko, V.I.

doi:10.1007/s11071-020-06015-5

Cited by 25 publications

(20 citation statements)

References 51 publications

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“…Distortion of square lattice makes in possible to obtain a tunable band gap in the phonon spectrum [47]. This study is a continuation of the work on DNVMs in a triangular lattice [11], but here we describe one-component DNVMs in a square β-FPUT lattice. All DNVMs described in in this work exist in any square lattice because they are determined only by lattice symmetry.…”

Section: Introductionmentioning

confidence: 88%

“…In [7][8][9], a method was developed for finding exact delocalized vibrational solutions for nonlinear lattices, based on the analysis of the lattice symmetry. Such solutions were referred to in original papers as bushes of nonlinear normal modes (BNNMs), and in some later studies as delocalized nonlinear vibrational modes (DNVMs) [10][11][12]. In particular, the BNNM theory was used to obtain DBs in a scalar nonlinear square lattice [13].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

One-component delocalized nonlinear vibrational modes of square lattices

Ryabov¹,

Chechin²,

Naumov

et al. 2022

Preprint

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All possible one-component delocalized nonlinear vibrational modes (DNVMs) in a square lattice are analyzed. DNVMs are obtained taking into account exclusively the symmetry of a square lattice, and therefore they exist regardless of the type of interactions between particles. In this work, the interactions of the nearest and next nearest neighbors are described by the β-FPU potential. For each DNVM, frequency, kinetic and potential energies are described as functions of amplitude. The mechanical stresses caused by DNVMs and the effect of DNVMs on the stiffness constants of the lattice are presented. DNVMs with higher vibration frequencies have a stronger effect on the mechanical properties of the lattice. Examples of analytical analysis of DNVMs are given. It is found that two of the sixteen one-component DNVMs can have frequencies above the phonon band in the entire range of their amplitudes. These DNVMs can be used to construct discrete breathers by applying localizing functions.

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Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

One-component delocalized nonlinear vibrational modes of square lattices

Ryabov¹,

Chechin²,

Naumov

et al. 2022

Preprint

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“…The bush theory essentially uses the apparatus of irreducible representations of point and space groups and can be applied for studies of the nonlinear dynamics of complex spatial structures. In particular, it was used in a large series of publications [14][15][16][17][18][19][20][21] devoted to the study of atomic vibrations in different molecular and crystal structures.…”

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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“…In the present study, we attempt to describe all possible DBs in a triangular β-FPUT lattice (named after Fermi, Pasta, Ulam, and Tsingou). The DBs are found by imposing localizing functions on delocalized nonlinear vibrational modes (DNVMs) of the triangular lattice [94] and linear chain [95]. Note that DNVMs are spatially periodic and exact oscillatory solutions to nonlinear equations of particle motion.…”

Section: Introductionmentioning

confidence: 99%

“…Consequently, the one-component DNVM is time periodic, while the m-component DNVM is, generally speaking, aperiodic with m incommensurate basis frequencies. For particular relations between amplitudes of DNVM components, periodic motion can be achieved [94,100].…”

Section: Introductionmentioning

confidence: 99%