Delocalized nonlinear vibrational modes of triangular lattices
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ORIGINAL PAPER
Delocalized nonlinear vibrational modes of triangular lattices Denis S. Ryabov · George M. Chechin · Abhisek Upadhyaya · Elena A. Korznikova Vladimir I. Dubinko · Sergey V. Dmitriev
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Received: 23 June 2020 / Accepted: 10 October 2020 © Springer Nature B.V. 2020
Abstract All possible one- and two-component delocalized nonlinear vibrational modes (DNVMs) in triangular lattice are analyzed. DNVMs are obtained considering solely upon the symmetry of the triangular lattice, and thus, they exist regardless the type of interactions between particles. In this work, the nearest-neighbor D. S. Ryabov · G. M. Chechin Southern Federal University, Institute of Physics, Stachki Ave. 194, Rostov-on-Don, Russia 344090 e-mail: [email protected] G. M. Chechin e-mail: [email protected] A. Upadhyaya Indian Institute of Technology Kharagpur, Kharagpur 721302, India e-mail: [email protected] E. A. Korznikova (B)· S. V. Dmitriev Institute for Metals Superplasticity Problems, Russian Academy of Sciences, Ufa, Russia 450001 e-mail: [email protected] E. A. Korznikova Institute of Molecule and Crystal Physics, Ufa Federal Research Centre of Russian Academy of Sciences, Oktyabrya Ave. 151, Ufa, Russia 450075 V. I. Dubinko NSC Kharkov Institute of Physics and Technology, Kharkiv 61108, Ukraine e-mail: [email protected] S. V. Dmitriev Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of Russian Academy of Sciences, Ufa, Russia 450000 e-mail: [email protected]
inter-particle interactions are described by the β-FPU potential. The one-component DNVMs are periodic in time, while the two-component ones are superposition of the two vibrational modes with, generally speaking, incommensurate frequencies. For many (but not for all) two-component DNVMs, time-periodic solutions can be obtained by a proper choice of the amplitudes of the two constitutive modes. For each DNVM, frequency and energy per particle are reported as the functions of vibration amplitude together with other characteristics. Keywords Triangular lattice · Nonlinear dynamics · Delocalized nonlinear vibrational mode · Exact solution
1 Introduction In recent decades, there has been growing interest in studying nonlinear particle vibrations in periodic structures, especially in crystal lattices. One type of such oscillations is spatially localized discrete breathers (or intrinsic localized modes) [1–6]. Here, we study the vibrations of a nonlinear lattice, which, in contrast to discrete breathers, are delocalized dynamic objects. There are no general methods for finding exact solutions describing particle vibrations in nonlinear lattices. Some of such solutions can be found by considering lattice space group symmetry. A theoretical approach for derivation of the lattice symmetry dictated exact delocalized nonlinear vibrational modes (DNVMs), often
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called as bushes of nonlinear normal modes (BNNMs), has been developed in the works [7–9]. DNVMs are periodic in space. O
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