Nonlinear Dynamics of a Linear Dust Particle Chain
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Nonlinear Dynamics of a Linear Dust Particle Chain A. M. Ignatov* Prokhorov General Physics Institute of the Russian Academy of Sciences, Moscow, 119991 Russia *e-mail: [email protected] Received March 16, 2020; revised April 14, 2020; accepted April 20, 2020
Abstract―The nonlinear stage of development of the dust particle chain instabilities is studied. It is assumed that dust particles interact with the plasma consisting of Maxwellian electrons and a directed flow of cold ions. The truncated equations are derived describing the nonlinear dynamics of the standing wave in the range of parameters close to the threshold for the development of the instability of coupled waves. It is shown that, in a narrow range of the ion flow Mach numbers, the instability of coupled waves becomes saturated in the weakly nonlinear stage of its development. In the remaining cases, the instability development becomes explosive. The nonlinear dynamics is also considered of the aperiodic period-doubling instability, which always becomes saturated in the weakly nonlinear stage of its development. Keywords: dusty plasma, plasma crystal DOI: 10.1134/S1063780X20090044
1. INTRODUCTION Dust particles, levitating above the horizontal electrode in the gas discharge plasma, often form various structures. The results from studying such formations are presented in many reviews and monographs (i.e., [1–4]). In particular, under certain conditions, the linear chains of particles can form, which, in equilibrium, are lined up in one straight line [5]. When analyzing the dynamics of the linear or more complex dust structures, it is important that, unlike the case of the classical mechanics, due to the presence of the directed ion flow, the third Newton law is not satisfied for the interparticle forces. The analytical form of such nonreciprocal forces is unknown; therefore, in the theoretical studies of the structure vibrations and stability, some model potentials are used. In particular, in the framework of the ion focus model, it was shown that the nonreciprocity of forces results in the coupling of the vertical and horizontal vibrations of the linear particle chain and development of the instability of coupled waves [6–8]. In [9], when analyzing the stability of the linear particle chain, the interparticle interaction was calculated numerically based on the model of plasma consisting of the cold ion flow and the Maxwellian electrons. It was shown that, along with the instability of coupled waves, in a certain range of plasma parameters, the aperiodic instability can also develop in such plasma. The goal of this work is to derive the truncated equations describing the nonlinear dynamics of a chain of particles in the range of parameters close to the threshold for the development of these instabili-
ties. The standard slowly varying amplitude approximation is used (for example, [10, 11]). In Section 2, the plasma model used is described and the dimensionless variables are introduced. Further (Section 3), the general equations of motion are pres
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