Wave Processes, Magnetic Fields and Nonlinear Structures in Dusty Plasma in the Vicinity of the Moon/Волновые процессы, магнитные поля и нелинейные структуры в пылевой плазме в окрестностях Луны тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Кассем Аттиа Ибрагем Кассем

• Кассем Аттиа Ибрагем Кассем
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Кассем Аттиа Ибрагем Кассем

Contents

Acknowledgement

List of Abbreviations and Symbols

Introduction 9 Chapter 1: Dusty plasmas in nature and characteristics of dusty

plasmas above the Moon

1.1 Dusty plasmas: history and laboratory

1.2 Dusty plasmas in space

1.3 Dusty plasma characteristics

1.4 Dusty plasmas above the Moon

1.4.1 Sources

1.4.2 Dust grain and lunar surface charging

1.4.3 Near-Moon Magnetic field

1.4.4 Plasma waves around the Moon

1.4.5 New missions

Chapter 2: Dust dynamics in the lunar dusty plasmas: Effects of

magnetic fields and dust charge variations

2.1 Introductory remarks

2.2 Model

2.3 First approximation

2.4 Effect of the magnetic field of the Earth's magnetotail

2.5 The magnetic field in the regions of magnetic anomalies of

the Moon

2.6 Discussion

Chapter 3: Lower-hybrid turbulence in the near-surface lunar

dusty plasmas

3.1 Introductory remarks

3.2 Lunar dusty plasma parameters

3.3 Lower-hybrid turbulence

3.4 Hydrodynamic instability

3.5 Nonlinear processes

3.6 Discussion

Chapter 4: Nonlinear dust acoustic solitary waves in a magnetized dusty plasma of the lunar exosphere

4.1 Modified Kadomtsev-Petviashvili equation for description of

nonlinear waves in dusty plasma

4.1.1 Mathematical model

4.1.2 Modified Kadomtsev-Petviashvili equation

4.1.3 Soliton solutions and their stability

4.2 Modified Zakharov- Kuznetsov equation in a magnetized

dusty plasma

4.2.1 Fundamental equations

4.2.2 Modified Zakharov-Kuznetsov equation

4.2.3 Solitary wave solution of the modified Zakharov-Kuznetsov equation

4.2.4 Instability analysis

4.3 Discussion

Conclusions

Bibliography

List of figures

List of tables

Введение диссертации (часть автореферата) на тему «Wave Processes, Magnetic Fields and Nonlinear Structures in Dusty Plasma in the Vicinity of the Moon/Волновые процессы, магнитные поля и нелинейные структуры в пылевой плазме в окрестностях Луны»

Introduction

Dust particles are considered to be the main significant components in cosmic plasma system. Perhaps there is only one exception to this rule, the Sun itself and the region in its immediate vicinity, where, due to high temperatures, dust cannot exist. Nano- and microscale dust particles are found in various locations throughout the solar system. These particles are present in interplanetary space, as well as in the plasma of the ionospheres and magnetospheres of planets. Additionally, they can be found in planetary rings and in the vicinity of cosmic bodies that lack their own atmosphere, such as Mercury, asteroids, and comets, etc. Furthermore, the study of dust particles in space dusty plasma systems is a rapidly growing field of research [1-8]. In reality, the charge on the dust grain is considered an important feature that occurs as a result of a number of processes, among them the electron and ion currents flowing into or out of the dust grain, as well as other processes like secondary emission, photo-emission of electrons, etc. These processes lead to dust charge fluctuations. The presence of massive (compared to ions and electrons) charged dust grain varies the characteristic of both spatial and temporal scales in plasma, and in some situations even gives rise to a new physics of certain phenomena. The mass of an individual isolated dust grain is about 106 — 1012 times the ion mass, and hence the mass of dusty plasma is mostly contained in the dust grains. Moreover, the presence of charged dust significantly affects collective processes, i.e., processes associated with the presence of oscillations or noises of finite amplitude (whose interaction with particles noticeably affects the macroscopic properties of plasma), self-organization processes in the medium, and also its dissipative properties. Furthermore, the presence of such fairly massive

(depending on the current plasma parameters) charged dust grains modifies the spectra of waves propagating in the medium, determines the nature of the development of instabilities and nonlinear processes, and so on.

In recent years, there has been a significant increase in interest in the direct study of cosmic dust particles. The NASA's Stardust mission was undertaken to collect and deliver to Earth particles from the vicinity of the nucleus of comet 81P / Wild 2. The project was successfully implemented on January 15, 2006, when a capsule with the samples of cometary matter returned to Earth [9]. In addition, the development of the theoretical approaches to describe cosmic dusty plasmas increased exponentially in the late 1990s [6]. One of the most significant natural dusty plasma systems in our solar system is the lunar dusty plasma system. In the recent American mission LADEE ("Lunar Atmosphere and Dust Environment Explorer") [10], lunar dust was studied using observations from orbit. The Russian Luna-25 and Luna-27 missions are being prepared to investigate the properties of dusty plasma above the LS [11]. The exploration of the lunar exosphere and the dynamics of the dust particles above the lunar surface has attracted a great deal of attention. That is why, in our dissertation, while discussing the properties of dust/dusty plasma in the space around the Sun, we will focus on its state above the LS. The study of the formation, properties, transport and dynamics of the charged dust grains in a dusty plasma system above the SSM under the influence of the magnetic field of the Earth's magnetotail is considered the main problem of the dissertation. Several theoretical studies on the dusty plasma transport and dynamics above the Moon have been performed by many authors not taking into account magnetic fields; for instance, in 1998, Nitter et al. [12], studied the levitation and the dynamics of charging dust particles in the photoelec-tron sheath above bodies in space. In 2008, Wang et al. [13], investigated the electrostatic levitation of DG on the lunar surface and showed that the

dust levitation condition in the terminator region is sensitively influenced by the ambient plasma condition and surface charging. In 2013, the size and elevation distributions of the charged DGs located over SSM were calculated by Popel et al. [14] for different angles between the local normal and the direction to the Sun. However, a theoretical model that provides a self-consistent description of the concentrations of photoelectrons and DGs located over SSM was presented and the effect of the magnetic field was not taken into account. The model [11, 15] of the electrostatically produced dusty plasmas does not take into account also nonstationary processes associated with the finiteness (in time) of daylight hours on the Moon. Another kind of nonstationary processes is related to the dust grain charge variations [16, 17], which can result in anomalous dissipation playing often the decisive role in dusty plasmas. The importance of the processes related to the dust grain charge variations in the electrostatically produced dusty plasmas at the Moon has not yet been clarified. Li et al. [18] showed that submicron sized dust particles can be levitated by the electric field above the Moon's surface. Moreover, the size and the charge of stably levitated dust particles are dependent on ambient plasma conditions. The distribution of DGs above the lunar surface with heights was proposed in [19], where the electrostatically ejected dust population can exist in the near-surface layer over the Moon, whereas the dust appearing in the lunar exosphere owing to the impacts of meteoroids is present everywhere.

Besides, linear and nonlinear waves research into dusty plasma system above the Moon was preformed in the absence of the magnetic field. Izvekova et al. [20] studied the turbulence of ion-acoustic waves and dust acoustic waves above the Moon. They showed that the ion acoustic waves are excited in some regions of the magnetosphere due to the development of a linear hydrodynamic instability, and the DAWs are generated in the entire region of the magnetotail interaction with the dusty plasma near the

Moon. In 2021, Mishra [21] explored the possibility of low-frequency dust wave excitation in the lunar dusty plasma environment. Kopnin and Popel [22] demonstrated the possibility of the existence and propagation of dust acoustic solitons in the dusty plasma of the lunar exosphere.

In reality, magnetic fields can exist. Approximately one-fourth of the lunar orbit passes through the Earth's magnetotail, which contains very rarefied plasmas in the geomagnetic tail wings as well as hotter and denser plasmas in the plasma sheath. However, studies examining this trend are scarce. In addition, there are the so-called areas of magnetic anomalies on the Moon associated with magnetic matter in the lunar crust. The measurements of near-surface magnetic fields on the visible side of the Moon, performed as part of the Apollo 12, 14, 15, and 16 missions, amounted to 3.8 x 10-4, 1.03 x 10-3, 3 x 10-5, and 3.27 x 10-3 G, respectively [23]. From satellite measurements [24], it is known that the largest (for the Moon) magnetic fields are present on the invisible side of the Moon. For this reason, we are also interested in studying the effect of these magnetic anomalies on the properties and the dynamics of dust grains above the Moon.

As a result, the main motivation in this work is to investigate the dynamics of dust grains under the influence of the magnetic field of the Earth's magnetotail above the sunlit side of the Moon (SSM), as well as to clarify the importance of processes related to dust grain charge variations in electrostatically produced dusty plasmas. In addition to studying the properties and features of linear and nonlinear wave processes and their stability above SSM. Furthermore, this work provides a developing macroscopic picture of the dust grain dynamics and dusty plasma above the Moon, which may be useful for future lunar missions, particularly Roscosmos missions (like Luna-25, Luna-26, and Luna-27) which will include experimental research on lunar dust and dusty plasma both near the LS (Luna-25 and Luna-27) and in orbit (Luna-26) around the Moon.

• The objectives of the dissertation

The purpose of the dissertation work is to build a developed picture of the influence of magnetic fields on dusty plasma in the vicinity of the Moon, including wave processes and nonlinear structures.

To achieve this aim, we find it necessary to implement more specific tasks:

- Describe and model the effect of the Earth's magnetotail magnetic fields and lunar magnetic anomalies on the dynamics of charged dust particles above the Moon.

- Investigate the trajectories of dust grains above the lunar surface in the case of the processes of variation in the charges of dust grains and their oscillations during the lunar day.

- Investigate wave processes of the lower-hybrid turbulence and the development of the lower-hybrid turbulence excitation which is due to the interaction of the Earth's magnetotail with the lunar dusty plasma.

- Derive and analyze nonlinear partial differential equations such as the modified Kadomtsev-Petviashvili equation and the modified Zakharov-Kuznetsov equation, which characterize the nonlinear dynamics of wave structures in dusty plasma above the Moon's illuminated side. The modified Kadomtsev-Petviashvili equation describes the nonlinear dynamics of wave structures in dusty plasma above the Moon's lighted side in the situation in which localization along the magnetic-field vector is much stronger than along other directions. Furthermore, the modified Zakharov-Kuznetsov equation describes the dynamics of nonlinear wave structures in the dusty plasma above the Moon's lighted

side in the case of low frequencies and the pancake-like shape of the wave packet in the direction along the external magnetic field.

• Scientific Novelty

- For the first time, the influence of magnetic fields of the Earth's magnetotail and of the magnetic fields in the regions of magnetic anomalies of the Moon on the transport of dust above the lunar surface and on the formation of dusty plasma above the Moon is performed. A new qualitative effect that does not exist in the absence of the magnetic field is provided. The magnetic component of the Lorenz force transports the dust grains from the region of high lunar latitudes toward the equator of the Moon, which is due to the magnetic fields of the Earth's magnetotail.

- For the first time, the investigation of the dust grain trajectories above the lunar surface shows that the processes of variation in the charges of dust grains, leading to the attenuation of their oscillations over the lunar surface, are too fast in comparison with the day duration on the Moon. In this connection, most of the dust in the dusty plasmas above above the Moon's lighted side can be considered as levitated.

- For the first time, analysis of linear and nonlinear processes is performed for the lower hybrid waves, which can be important in the dusty plasma above the Moon. It is shown that the lower-hybrid turbulence can be generated wherever the Earth's mag-netotail interacts with the near-surface dusty plasmas above the Moon. The electric fields generated by the development of lower-hybrid turbulence are evaluated.

- For the first time, in application to the lunar dusty plasma, a nonlinear modified Kadomtsev-Petviashvili equation is obtained to describe two-dimensional nonlinear dust acoustic structures above the Moon. Stability analysis of the one-dimensional soliton solution showed that it is stable.

- For the first time, the modified Zakharov-Kuznetsov equation is derived to describe the dynamics of nonlinear structures in the dusty plasma above the Moon in the case of low frequencies and pancake-like shape of the wave packet in the direction along the external magnetic field above the Moon's lighted side.

• Scientific Statements for the defense

- The physical and mathematical model is developed to describe the dynamics of charged dust particles above the Moon under the effect of the magnetic fields of the Earth's magnetotail and the lunar magnetic anomalies. It is shown that due to the action of the magnetic fields, transfer of charged dust particles is possible over long distances above the Moon's surface, and the dusty plasma above the day-side of the Moon can exist in the entire range of lunar latitudes.

- It is shown that most of the dust particles in dusty plasma above the Moon's lighted side are levitated. Only very fine particles (with sizes smaller than several dozens of ^m) can be treated as lofted. This is, in particular, related to a fast process of charge variation of dust particles in comparison with the day duration on the Moon.

- It is shown that when the Earth's magnetotail interacts with dusty plasmas near the lunar surface, the excitation of lower-

hybrid turbulence is possible. The effective collision frequency, which determines the loss of ion momentum as a consequence of the interaction between magnetotail ions and lower-hybrid waves, is determined. The typical electric fields are estimated, which appear due to lower-hybrid turbulence development. Such electric fields can make a significant contribution to the total electric field above the lunar surface, which should be considered in further experimental research on electric fields above the Moon.

- A two-dimensional description of nonlinear waves in dusty plasmas above the Moon is given. In this connection, the modified Kadomtsev-Petviashvili and Zakharov-Kuznetsov equations are derived under the conditions of the lunar dusty plasmas and analyzed. Soliton solutions of these equations are obtained. It is shown that the one-dimensional soliton solutions of these equations are stable.

• Scientific and practical significance of the work

The dissertation results can be used by a wide range of specialists involved in the study of physical characteristics of dusty plasma, as well as contribute to further theoretical and practical research on dusty plasma physics above the Moon or in space.

The dynamics of the dust grains above the lunar surface and the processes of charge variations of dust grains under the influence of the magnetic fields of the Earth's magnetotail and magnetic anomalies may be helpful in future lunar missions. For instance, the future Roscosmos missions, Luna-25, Luna-26, and Luna-27, will include experimental research on lunar dust and dusty plasma both near the lunar surface (Luna-25 and Luna-27) and at the orbit (Luna-26) around

the Moon.

The solitary wave study above the Moon can be contributed to interpretation from the point of view of the description of the so-called transient lunar phenomena representing short-lived light, changes in color or appearance on the surface of the Moon. Phenomena of this kind are sometimes related to the sporadic release of gases on the Moon. However, it seems feasible to suggest that upon propagation of electrostatic solitons along the lunar surface, glow can be caused by plasma emission from the region of soliton localization, i.e., solitons can contribute to the explanation of these or other transient lunar phenomena.

• Approbation of the work

The main results of the dissertation were presented and discussed at seminar at the Space Research Institute (IKI, RAS), as well as 9 oral and poster presentations at international conferences, international workshops and Russian conferences. The list of scientific conferences and seminars on the dissertation topic is given as follows:

[1] Izvekova Yu. N., Kassem A. I., Popel S. I., Morozova T. I., Zelenyi L. M. Dusty plasmas at the Moon: Effects of magnetic fields. In XXXV International Conference on Equations of State for Matter (Elbrus, Russia), 280, 1-6 March 2020.

[2] Kassem, A. I., Popel, S. I., Izvekova, Y. N., and Zelenyi, L.

M. Lower-hybrid waves in the exosphere of the Moon. In XVII Conference of Young Scientists "Fundamental and applied space research" (IKI RAS, Moscow, Russia), 49-60, 30 September-2 October 2020.

[3] Popel S. I., Golub' A. P., Kassem A. I. and Zelenyi L. M. Dust dynamics at the Moon: levitation or lofting. In The Twelfth Moscow Solar System Symposium (IKIRAS, Moscow, Russia), 1115 October 2021.

[4] Kassem A. I., Popel S. I., Golub' A. P., Zelenyi L. M. On the

influence of the Earth's magnetic field on the dynamics of dust particles in the exosphere of the Moon. In 64th International MIPT Scientific Conference 2021, (MIPT, Moscow region, Dolgo-prudny, Russia), 29 November-3 December 2021.

[5] Kassem A. I. participated. In the first international conference on space education: THE ROAD TO SPACE, (IKI RAS, Moscow, Russia), 5-8 October, 2021.

[6] Kassem A. I., Kopnin S. I., Popel S. I., Zelenyi L. M. Characteristic behavior of dust acoustic solitons in the dusty plasma of the lunar exosphere. In 18th International workshop complex systems of charged particles and their interactions with electromagnetic radiation, (GPIRAS, Moscow, Russia), 11-13 April 2022.

[7] Kassem A. I., Popel S. I., Golub' A. P., Zelenyi L. M. The impact of the lunar magnetic anomalies and Earth's magnetic field on the dust movements in the lunar dusty plasma. In 9th International Conference on the Physics of Dusty Plasmas ICPDP 2022, (IKIRAS, Moscow, Russia), 23-27 May 2022.

[8] Kassem A. I., Popel S. I., Kopnin S. I., Zelenyi L. M. Modified Zakharov-Kuznetsov equation for description of nonlinear perturbations in plasma of dusty lunar exosphere. In 9th ICPDP 2022, (IKIRAS, Moscow, Russia), 23-27 May 2022.

[9] Kassem A. I., Kopnin S. I., Popel S. I., Zelenyi L. M. Modified Zakharov-Kuznetsov equation for description of nonlinear waves in magnetized dusty plasmas of the lunar exosphere. In 19th International workshop complex systems of charged particles and their interactions with electromagnetic radiation, (GPI RAS, Moscow, Russia), 10-13 April 2023.

• List of publications

The dissertation results are published in 6 papers indexed in Scopus and Web of Science and summarized as the following:

[1] Popel, S. I., Kassem, A. I., Izvekova, Y. N., and Zelenyi, L.

M. Lower-hybrid turbulence in the near-surface lunar dusty plasmas. Physics Letters A, VOL 384(26), 126627, 2020.

[2] Izvekova, Y. N., Kassem, A. I., Popel, S. I., Morozova, T. I., and Zelenyi, L. M. Dusty plasmas at the Moon: Effects of magnetic fields. Journal of Physics: Conference Series, VOL 1787(1), 012051, 2021.

[3] Popel, S. I., Golub', A. P., Kassem, A. I., and Zelenyi, L. M.

Dust dynamics in the lunar dusty plasmas: Effects of magnetic fields and dust charge variations. Physics of Plasmas, VOL 29(1), 013701, 2022.

[4] Kassem A. I., Kopnin S. I., Popel S. I., Zelenyi, L. M. Modified Kadomtsev-Petviashvili Equation for the Description of Nonlinear Disturbances in the Plasma of the Dusty Exosphere of the Moon. Plasma Physics Reports, VOL 48(4), 361-366, 2022.

[5] Popel, S. I., Golub', A. P., Kassem, A. I., and Zelenyi, L. M.

On the role of magnetic fields in the plasma of dusty lunar exo-sphere. Plasma Physics Reports, Vol 48(5), 512-517, 2022.

[6] Kassem A. I., Kopnin S. I., Popel S. I., Zelenyi, L. M. Modified Zakharov-Kuznetsov Equation for Description of Low-Frequency Nonlinear Perturbations in the Plasma of the Dusty Exosphere of the Moon. Plasma Physics Reports, VOL 48(9), 1005-1012, 2022.

• Further related publications

[1] El-Wakil, S. A., Abulwafa, E. M., El-Hanbaly, A. M., and Kassem, A. I. Propagation of Solitary Waves and Double-Layers in Electron-Positron Pair Plasmas with Stationary Ions and Nonextensive Electrons. In International Journal of Applied and Computational Mathematics, VOL. 5(4), 1-23, 2019.

• Author's Contribution

The author participated in the statement of the problems, performed analytical and numerical calculations and mainly participated in formulation of the statement of conclusions.

• The structure of the thesis

The thesis is divided into an introduction, four chapters, a conclusion and a list of 135 references. The dissertation volume is 123 pages, includes 25 figures and one table.

• Organization of the thesis

Chapter 1 starts by giving some introductory facts about dust particles and is followed by a scientific literature review on the observation and formation of dust particles in space. In addition, this chapter discusses the

sources and creation of dust particles as well as their charging above the Moon. The significance of studying the influence of the magnetic fields of the Earth's magnetotail and lunar magnetic anomalies on the dynamics of the DGs above the Moon and waves in dusty plasma is presented in this chapter.

In chapter 2, we investigate the influence of the magnetic fields of the Earth's magnetotail and magnetic anomalies of the lunar surface on the formation of the electrostatically produced DP near the Moon and study the trajectories of dust grains above the lunar surface in the case of the processes of variation in the charges of dust grains and their oscillations during the Moon day.

Chapter 3 investigates the possibility of lower-hybrid turbulence excitation in dusty plasma on the Moon. The effective collision frequency, which determines the loss of ion momentum as a consequence of the interaction between magnetotail ions and lower-hybrid waves, is determined.

In chapter 4, nonlinear equations such as the modified Kadomtsev-Petviashvili equation and the modified Zakharov-Kuznetsov equation characterizing non-linear dynamics of wave structures in dusty plasma above SSM are derived and analyzed. In addition, we discuss the stability of a one-dimensional soliton solution of the modified Kadomtsev-Petviashvili equation and the modified Zakharov-Kuznetsov equation.

Finally, in conclusion, we formulate the thesis's general conclusions.

Заключение диссертации по теме «Другие cпециальности», Кассем Аттиа Ибрагем Кассем

Conclusions

The main results of the dissertation work are concluded as follows:

1. Investigation of the influence of magnetic fields of the Earth's mag-netotail and of the magnetic fields in the regions of magnetic anomalies of the Moon on the dust transport and on the formation of dusty plasmas above the Moon has been performed. The research had shown a possibility of the existence of positively charged dust for the whole range of the lunar latitudes (from — 90o to 90o). Consequently, dusty plasmas above the sunlit part of the Moon can exist in the entire range of lunar latitudes. Dust grain transport from the region of high lunar latitudes toward the equator of the Moon has been demonstrated to be due to the uncompensated magnetic part of the Lorentz force, which is due to the magnetic fields of the Earth's magnetotail. It provides a new qualitative effect that does not exist in the absence of the magnetic field. The magnetic component of the Lorenz force acting on the dust grain in the fields of magnetic anomalies is either lower or comparable to the similar force calculated for the magnetic field of the Earth's magnetotail in lunar orbit. Nevertheless, due to the existing localization of the regions of magnetic anomalies, these regions can lead only to some changes in dust grain trajectories, deflecting them from those formed as a result of the influence of magnetic fields of the Earth's magnetotail, the general trend of the motion of dust grains remaining the same as in the case of the dust grain motion under the action of the magnetic fields of the Earth's magnetotail.

2. The investigation of the dust grain trajectories above the lunar surface has shown that the processes of variation in the charges of dust grains, leading to the attenuation of their oscillations over the lunar surface, are too fast in comparison with the day duration on the Moon. In this connection, most of the dusts in the dusty plasmas above the sunlit part of the lunar surface can be considered as levitated. Only very fine grains (with sizes smaller than several dozens of ^m) do not go into levitating grain mode during the entire day on the Moon and can be treated as lofted.

3. An analysis of linear and nonlinear processes related to the presence of magnetic fields, which can be important in dusty plasmas at the Moon has been performed. It has been shown the significance of lower-hybrid turbulence excitation which is due to the interaction of the Earth's magne-totail with dusty plasma near the lunar surface. The growth rates characterizing the excitation of the lower-hybrid turbulence have been found. It has been demonstrated that in contrast to the situation with ion-acoustic turbulence, which is excited only in the transient and/or boundary mag-netospheric layers, the lower-hybrid turbulence can be generated wherever the Earth's magnetotail interacts with the near-surface dusty plasmas at the Moon. The effective collision frequency, which determines the loss of ion momentum as a consequence of the interaction between magnetotail ions and lower-hybrid waves, has been determined, and the typical electric fields, which appear due to lower-hybrid turbulence development, have been estimated.

4. Nonlinear equations such as the modified Kadomtsev-Petviashvili equation and the modified Zakharov-Kuznetsov equation characterizing nonlinear dynamics of wave structures in dusty plasma above the Moon's illuminated side have been derived and analyzed. The modified Kadomt-sev- Petviashvili equation describes nonlinear dynamics of nearly one-dimensional wave structures in dusty plasma above the Moon's lighted side in the situation in which localization along magnetic-field vector is much stronger than along other directions. The equation differs from the ordinary Kadomtsev-Petviashvili equation by the nonlinear term being non-analytical. Modified Kadomtsev-Petviashvili differs from generalizations of the Kadomtsev-Petviashvili equation in which nonlinearity retains the same form as in the ordinary Kadomtsev-Petviashvili equation but higher-order corrections for dispersion are taken into account. An analytical expression governing one-dimensional soliton solution to the modified Kadomt-sev-Petviashvili equation has been obtained. The solution differs from the well-known one-dimensional soliton solutions to the Korteweg-De Vries and ordinary Kadomtsev-Petviashvili equations. Stability analysis of the one-dimensional soliton solution showed that it is stable. The modified Zakharov-Kuznetsov equation is the nonlinear equation that describes the

dynamics of nonlinear wave structures in the dusty plasma above the illuminated surface of the Moon in the case of low frequencies and pancake-like shape of wave packet in the direction along the external magnetic field. The analytical formula for the one-dimensional soliton solution of the modified Zakharov-Kuznetsov equation has been derived. Possible applications of the obtained soliton solutions are related to the description of the so-called transient lunar phenomena representing short-lived light, changes in color or appearance on the surface of the Moon.

Список литературы диссертационного исследования кандидат наук Кассем Аттиа Ибрагем Кассем, 2023 год

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