Nonlinear periodic waves in a deformable medium modeled by chains of active Morse–van der Pol particles

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Abstract

Using numerical modeling methods, the processes of generation and propagation of nonlinear periodic waves in a deformable medium modeled by various chains of active Morse–van der Pol particles were studied. In a wide range of chain lengths, the intervals of change in wave periods are determined. It is shown that in short chains the conservative Morse forces are much greater than the spatially dependent forces of active friction, as a result of which the wave process occurs according to a conservative scenario. In long chains, the process of transformation of a nonlinear periodic wave into a dissipative soliton, the minimum speed of which corresponds to the maximum value of the period, has been revealed. It has been established that the dependence of the minimum period on the number of particles in the chain is almost linear. The instability of the propagation of initial disturbances consisting of several previously identified identical periodic solutions is demonstrated.

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About the authors

А. I. Zemlyanukhin

Yuri Gagarin State Technical University of Saratov

Author for correspondence.
Email: ispavlov@mail.ru
Russian Federation, Saratov

A. V. Bochkarev

Yuri Gagarin State Technical University of Saratov

Email: ispavlov@mail.ru
Russian Federation, Saratov

V. I. Erofeev

Mechanical Engineering Research Institute of the Russian Academy of Sciences; National Research Lobachevsky State University of Nizhny Novgorod

Email: ispavlov@mail.ru

Federal Research Center A.V. Gaponov-Grekhov Institute of Applied Physics of the RAS

Russian Federation, Nizhny Novgorod; Nizhny Novgorod

I. S. Pavlov

Mechanical Engineering Research Institute of the Russian Academy of Sciences; National Research Lobachevsky State University of Nizhny Novgorod

Email: ispavlov@mail.ru

Federal Research Center A.V. Gaponov-Grekhov Institute of Applied Physics of the RAS

Russian Federation, Nizhny Novgorod; Nizhny Novgorod

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Supplementary files

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2. Fig. 1. The shape of a periodic wave in a Morse–van der Pol circuit for N = 10.

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3. Fig. 2. (a) — Profiles of displacement and (b) — velocities of particles in a periodic wave at N = 13 for the minimum and maximum values ​​of the period.

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4. Fig. 3. Profiles of periodic waves corresponding to minimum periods for N = 13..15.

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5. Fig. 4. Lower and upper boundaries of wave periods in chains of N particles.

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6. Fig. 5. Profiles of periodic waves for (a) N = 7 and (b) N = 25.

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7. Fig. 6. (a) — Dependence u(t), found using procedures 1 and 2 for N = 15 and T = 3.3531; (b) — the same dependence, obtained by the second method.

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8. Fig. 7. The process of destruction of a periodic solution in a 2N-chain.

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