Exact travelling wave solutions in MHD and plasma physics
Creators: name: ALDHABANI, Musaad Sabih D
Publisher: name: RMIT University
Year: 2024
Subjects: classificationCode: 510602
subject: Plasma physics; fusion plasmas; electrical discharges

description: Over the last four decades, much attention has been paid to nonlinear studies on exact travelling wave solutions for some nonlinear physical systems such as ion-acoustic waves, incompressible magnetohydrodynamics (MHD) waves, and isothermal magnetostatic atmospheres. It is not surprising then that exact travelling solution-based techniques have become important in the study of nonlinear physical systems.
 
Nonlinear evolution equations are commonly used as models for explaining complex physical processes in different fields of sciences, especially in fluid mechanics, solid state physics, plasma physics and chemical physics. Given a nonlinear partial differential equation (NLPDE), there is no general way of knowing whether it has soliton solutions or not, or how the soliton solutions can be found. In order to gain a better understanding of the underlying phenomena as well as their further applications in practical life, it is important to seek their exact solutions.

Analytical solutions to NLPDEs play an important role in nonlinear science, especially in nonlinear physical science since they can provide more insight into the physical aspects of the problem and thus lead to further applications. Moreover, new exact solutions may help researchers to find new phenomena [1-11]. In various science and engineering fields, nonlinear wave phenomena arise, such as in fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical physics and geochemistry. In the theory of nonlinear wave dispersion, dissipation, diffusion, reaction and convection, nonlinear wave phenomena are very common [12-22].

This thesis consists of a summary and seven chapters as detailed below.

Chapters 1, 2 and 3 are Introductory chapters dealing with the historical discussions of solitons and travelling wave solutions. This is followed by a brief survey of how mathematicians and physicists began to work on certain problems of mutual interest. This is followed by a brief survey of MHD and ion-acoustic waves. In addition these chapters provide the background material used in this thesis. They covers the fundamental concepts of known results concerning our works to make this thesis somewhat self-contained.

Chapters 4,5 and 6 are primarily focused on finding travelling wave solutions in certain plasma physics problems. In chapter 4 we address acoustic waves of ions, which are the variation of sound waves in plasma physics. Such waves spread as more mobile electrons protect the electrical field from the ions and set up their own ¿eld to move the ions in a tenuous plasma. These waves could be excited, but there are infrequent collisions between the electrons and ions. In these waves the natural magnetic field is zero. However, the solutions in chapter 5 for a three-dimensional incompressible MHD system are obtained using the sine - cosine method and the Riccati auxiliary equation. This chapter obtains soliton solutions with the aid of the software Mathematica.

In chapter 6, the Bäcklund Transformations (BTs) method is proposed to look for exact solutions for the nonlinear differential equations resulting from solar MHD. The BTs methodology is applicable to search for exact solutions from magnetostatic equations in solar physics to the sinh-Poisson equations.

Finally, chapter 7 provides a brief conclusion and considerations for possible future work in these areas.