Resonance regions of extended Mathieu equation
One of the mechanisms of energy transfer between degrees of freedom of dusty plasma system is based on parametric resonance. Initial stage of this process can de described by equation similar to Mathieu equation. Such equation is studied by analytical and numerical approach. The numerical solution of the extended Mathieu equation is obtained for a wide range of parameter values. Boundaries of resonance regions, growth rates of amplitudes and times of onset are obtained. The energy transfer between the degrees of freedom of dusty plasma system can occur over a wide range of frequencies.
Wave processes occurring under the interaction of the Earth's magnetosphere with dusty plasma near the lunar surface are studied. Ion-acoustic waves are shown to be excited in some regions of the magnetosphere due to the development of a linear hydrodynamic instability. This results in the excitation of ion-acoustic turbulence in these regions. Dust-acoustic waves are demonstrated to be generated due to the development of linear kinetic instability in the entire region of magnetotail interaction with dusty plasma near the Moon. Correspondingly, dust-acoustic turbulence can be excited in the entire region of the interaction of the Earth's magnetosphere with dusty plasma near the lunar surface. We discuss magnetic reconnection processes, which are related to the development of plasma turbulence at the Moon.
The self-consistency and basic openness of dusty plasma, charge fluctuations, high dissipation and other features of dusty plasma system lead to the appearance of a number of unusual and unique properties of dusty plasma. “Anomalous” heating of dusty particles, anisotropy of temperatures and other features, parametric resonance, charge fluctuations and interaction potential are among these unique properties. Study is based on analytical approach and numerical simulation. Mechanisms of “anomalous” heating and energy transfer are proposed. Influence of charge fluctuations on the system properties is discussed. The self-consistent, many-particle, fluctuation and anisotropic interparticle interaction potential is studied for a significant range of gas temperature. These properties are interconnected and necessary for a full description of dusty plasmas physics.
A theory is developed which describes the processes of dust particle charging in the situation when dust particles are subjected to the action of a beam of electrons. It is shown that in this situation it is necessary to consider the electron field emission in addition to the influence of the electron beam on the dust particle. We calculate the current of the electron field emission modified by the Schottky effect and find the steady-state dust particle charge. We show that in the situation considered the electrostatic energy of the dust particle is much smaller than the electron energy in the beam.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.