Dynamical Systems and Nonlinear Waves in Plasmas

✍ Scribed by Santo Banerjee, Asit Saha

• Publisher: CRC Press
• Year: 2021
• Tongue: English
• Leaves: 218
• Edition: 1
• Category: Library

✦ Synopsis

Dynamical systems and Nonlinear Waves in Plasmas is written in a clear and comprehensible style to serve as a compact volume for advanced postgraduate students and researchers working in the areas of Applied Physics, Applied Mathematics, Dynamical Systems, Nonlinear waves in Plasmas or other nonlinear media.

It provides an introduction to the background of dynamical systems, waves, oscillations and plasmas. Basic concepts of dynamical systems and phase plane analysis for the study of dynamical properties of nonlinear waves in plasmas are presented. Different kinds of waves in plasmas are introduced. Reductive perturbative technique and its applications to derive different kinds of nonlinear evolution equations in plasmas are discussed. Analytical wave solutions of these nonlinear evolution equations are presented using the concept of bifurcation theory of planar dynamical systems in a very simple way. Bifurcations of both small and arbitrary amplitudes of various nonlinear acoustic waves in plasmas are presented using phase plots and time-series plots. Super nonlinear waves and its bifurcation behaviour are discussed for various plasma systems. Multiperiodic, quasiperiodic and chaotic motions of nonlinear plasma waves are discussed in presence of external periodic force. Multistability of plasma waves is investigated. Stable oscillation of plasma waves is also presented in dissipative plasmas.

The book is meant for undergraduate and postgraduate students studying plasma physics. It will also serve a reference to the researchers, scientists and faculties to pursue the dynamics of nonlinear waves and its properties in plasmas. It describes the concept of dynamical systems and is useful in understanding exciting features, such as solitary wave, periodic wave, supernonlinear wave, chaotic, quasiperiodic and coexisting structures of nonlinear waves in plasmas. The concepts and approaches, discussed in the book, will also help the students and professionals to study such features in other nonlinear media.

✦ Table of Contents

Cover
Title Page
Copyright Page
Dedication
Preface
Table of Contents
1. Introduction
1.1 Plasma as a state
1.2 Plasmas exist in nature
1.2.1 Ionosphere
1.2.2 Van Allen belts
1.2.3 Aurorae
1.2.4 Solar corona
1.2.5 Core of the sun
1.2.6 HII regions
1.3 Concept of temperature
1.3.1 Existence of several temperatures
1.3.2 Electron and ion temperatures
1.3.3 Quasineutrality in plasma
1.4 Debye length and Debye sphere
1.5 Criteria for plasma
1.6 Plasma frequency
1.7 Applications of plasma
1.7.1 Space physics
1.7.2 Astrophysics
1.7.3 Gas lasers
1.7.4 Industrial application
1.8 Fluid description of plasma
1.8.1 Maxwell’s equation
1.8.2 Equation of motion
1.8.3 Equation of continuity
1.8.4 Equation of state of plasma
1.8.5 Poisson equation
References
2. Dynamical Systems
2.1 Introduction to dynamical systems
2.1.1 One-dimensional system
2.1.1.1 Equilibrium point and its stability
2.1.1.2 Trajectory and phase portrait
2.1.1.3 Example
2.1.2 Linear stability analysis
2.1.2.1 Example
2.1.3 Potentials
2.1.3.1 Example
2.1.3.2 Example
2.1.4 Bifurcations
2.1.5 Linear system in two-dimension
2.1.5.1 Example
2.1.6 Phase plane analysis
2.1.6.1 Nonlinear system in two-dimension
2.1.6.2 Conservative system
2.1.6.3 Example
2.1.6.4 Example
2.1.6.5 Hamiltonian system
2.1.6.6 Example
References
3. Waves in Plasmas
3.1 Introduction to wave modes
3.1.1 Ion-acoustic (IA) waves
3.1.2 Dust-acoustic (DA) waves
3.1.3 Dust-ion-acoustic (DIA) waves
3.1.4 Upper hybrid wave
3.1.5 Electrostatic cyclotron waves
3.1.6 Lower hybrid wave
3.2 Reductive perturbation technique and evolution equations
3.2.1 The KdV equation
3.2.2 The Burgers equation
3.2.3 The KP equation
3.2.4 The ZK and mZK equations
3.3 Analytical wave solutions of evolution equations
3.3.1 Analytical wave solution of the KdV equation
3.3.2 Analytical wave solution of the mKdV equation
3.3.3 Analytical wave solution of the KP equation
3.3.4 Analytical wave solution of the mKP equation
3.3.5 Analytical wave solution of the ZK equation
3.3.6 Analytical wave solution of the mZK equation
3.3.7 Analytical wave solution of the Burgers equation
References
4. Bifurcation of Small Amplitude Waves in Plasmas
4.1 Introduction
4.2 Bifurcation of ion-acoustic waves with small amplitude
4.2.1 Basic equations
4.2.2 Derivation of the KdV equation
4.2.3 Formation of dynamical system
4.2.4 Phase plane analysis
4.2.5 Wave solutions
4.3 Bifurcation of dust-ion-acoustic waves with small amplitude
4.3.1 Governing equations
4.3.2 Derivation of the KP equation
4.3.3 Formation of dynamical system and phase portraits
4.3.4 Wave solutions
4.4 Bifurcation of dust-acoustic waves with small amplitude
4.4.1 Basic equations
4.4.2 Derivation of the Burgers equation
4.4.3 Formation of dynamical system and phase portraits
4.4.4 Wave solutions
4.5 Bifurcation of electron-acoustic waves with small amplitude
4.5.1 Basic equations
4.5.2 Derivation of the KdV equation
4.5.3 Formation of dynamical system and phase portraits
4.5.4 Wave solutions
References
5. Bifurcation of Arbitrary Amplitude Waves in Plasmas
5.1 Introduction
5.2 Bifurcation of ion-acoustic waves with arbitrary amplitude
5.2.1 Basic equations
5.2.2 Formation of dynamical system and phase portraits
5.2.3 Wave solutions
5.3 Bifurcation of dust-ion-acoustic waves with arbitrary amplitude
5.3.1 Basic equations
5.3.2 Formation of dynamical system and phase portraits
5.3.3 Wave solutions
5.4 Bifurcation of dust-acoustic waves with arbitrary amplitude
5.4.1 Basic equations
5.4.2 Formation of dynamical system and phase portraits
5.4.3 Wave solutions
5.5 Bifurcation of electron-acoustic waves with arbitrary amplitude
5.5.1 Basic equations
5.5.2 Formation of dynamical system and phase portraits
5.5.3 Wave solutions
References
6. Bifurcation Analysis of Supernonlinear Waves
6.1 Introduction: supernonlinear waves
6.1.1 Different kind of trajectories
6.2 Bifurcation of supernonlinear ion-acoustic waves
6.2.1 Basic equations
6.2.2 Modified KdV equation
6.2.3 Formation of dynamical system and phase portraits
6.2.4 Wave solutions
6.3 Bifurcation of supernonlinear dust-acoustic waves
6.3.1 Basic equations
6.3.2 Formation of dynamical system and phase portraits
6.3.3 Wave solutions
6.4 Bifurcation of supernonlinear electron-acoustic waves (EAWs)
6.4.1 Basic equations
6.4.2 The evolution equation and dynamical system
6.4.3 Wave solutions
References
7. Chaos, Multistability and Stable Oscillation in Plasmas
7.1 Chaos in a conservative dusty plasma
7.1.1 Basic equations
7.1.2 Multiperiodic, quasiperiodic and chaotic oscillations
7.2 Multistability of electron-acoustic waves
7.2.1 Basic equations
7.2.2 Multistability
7.3 Stable oscillation in a dissipative plasma
7.3.1 Model equations
7.3.2 The KdV-Burgers equation
7.3.3 Stability analysis of DAWs
References
Index