This book is written in a lucid and systematic way for advanced postgraduates and researchers studying applied mathematics, plasma physics, nonlinear differential equations, nonlinear optics, and other engineering branches where nonlinear wave phenomena is essential. In sequential order of the book's development, readers will understand basic plasmas with elementary definitions of magnetized and unmagnetized plasmas, plasma modeling, dusty plasma and quantum plasma. Following which, the book describes linear and nonlinear waves, solitons, shocks and other wave phenomena, while solutions to common nonlinear wave equations are derived via standard techniques. Readers are introduced to elementary perturbation and non-perturbation methods. They will discover several evolution equations in different plasma situations as well as the properties of solitons in those environments. Pertaining to those equations, readers will learn about their higher order corrections, as well as their different forms and solutions in non-planar geometry. The book offers further studies on different types of collisions between solitons in plasma environment, phenomena of soliton turbulence as a consequence of multi-soliton interactions, properties of large amplitude solitary waves which are discovered via non-perturbative Sagdeev's Pseudopotential Approach, as well as the speed and shape of solitons. Finally, the book reveals possible future developments of research in this rich field.
Author(s): Prasanta Chatterjee, Kaushik Roy, Uday Narayan Ghosh
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 378
City: Singapore
Contents
Preface
Acknowledgments
Chapter 1. Introduction to Plasmas
1.1 Introduction
1.2 Saha Equation and Plasma Temperature
1.3 Basic Concepts of Plasma
1.3.1 Basic dimensionless parameters
1.3.2 Debye length and Debye shielding
1.3.3 Quasineutrality
1.3.4 Response time
1.3.5 Plasma frequency
1.3.6 Collisions and coupling limit
1.4 Criteria for Plasma
1.5 High-Temperature Plasmas
1.6 Mathematical Description
1.7 Magnetized Plasmas
1.8 Single Particle Motion in Uniform Electric and Magnetic Field
1.9 Fluid Approach
1.10 Maxwell’s Equations
1.11 Electromagnetic Wave Equation in Free Space
1.12 Plasma Kinetic Theory
1.12.1 Distribution function
1.12.2 Macroscopic variables
1.12.3 Maxwellian distribution function
1.12.4 Non-Maxwellian distribution in plasmas
1.12.5 Nonthermal distribution
1.12.6 Superthermal distribution
1.12.7 q-nonextensive distribution
1.13 Closure Form of Moment Equation
1.13.1 Equation of continuity
1.13.2 Equation of motion
1.13.3 Equation of energy
1.14 Dusty Plasma
1.15 Quantum Plasma
1.16 Quantum Plasma Models
References
Chapter 2. Introduction to Waves in Plasma
2.1 Introduction
2.2 Mathematical Description of Waves
2.3 Dispersion Relation
2.4 Linear Waves in Plasmas
2.5 Plasma Oscillation
2.6 Electromagnetic Waves
2.7 Upper Hybrid Frequency
2.8 Electrostatic Ion Cyclotron Waves
2.9 Lower Hybrid Frequency
2.10 Electromagnetic Waves with B0 = 0
2.11 Electromagnetic Waves Perpendicular to B0
2.11.1 Ordinary wave
2.11.2 Extraordinary wave
2.12 Electromagnetic Waves Parallel to B0
2.13 Hydromagnetic Waves
2.13.1 Alfven wave
2.13.2 Magnetosonic wave
2.14 Some Acoustic Type of Waves in Plasmas
2.14.1 Electron plasma waves
2.14.2 Ion acoustic waves
2.14.3 Dust acoustic waves
2.14.4 Dust ion acoustic waves
2.15 Nonlinear Wave
2.16 Solitary Waves and Solitons
2.16.1 History of solitary waves and solitons
2.17 Properties of Solitons
References
Chapter 3. Solution of Nonlinear Wave Equations
3.1 Nonlinear Waves
3.2 Direct Method
3.2.1 Korteweg–de Vries (KdV) equation
3.2.2 Cnoidal waves
3.2.3 Modified KdV (MKdV) equation
3.2.4 Schamel-type KdV (S-KdV) equation
3.2.5 Burgers’ equation
3.2.6 KP equation
3.2.7 Modified KP equation
3.3 Hyperbolic Tangent Method
3.3.1 KdV equation
3.3.2 Modified KdV equation
3.3.3 Burgers’ equation
3.3.4 KdV Burgers’ equation
3.3.5 KP equation
3.4 Tanh–Coth Method
3.4.1 KdV equation
3.4.2 Burgers’ equation
3.5 Solution of KP Burger Equation
3.6 Conservation Laws and Integrals of the Motions
3.6.1 Conserved quantity of KdV equation
3.7 Approximate Analytical Solutions
3.7.1 Damped KdV equation
3.7.2 Force KdV equation
3.7.3 Damped-force KdV equation
3.8 Multisoliton and Hirota’s Direct Method
3.8.1 Hirota’s method
3.8.2 Multisoliton solution of the KdV equation
3.8.3 Multisoliton solution of the KP equation
References
Chapter 4. RPT and Some Evolution Equations
4.1 Perturbation Technique
4.2 Reductive Perturbation Technique
4.3 Korteweg–de Vries (KdV) Equation
4.4 Modified KdV (MKdV) Equation
4.5 Gardner’s Equation
4.6 Gardner and Modified Gardner’s (MG) Equation
4.7 Damped Forced KdV (DFKdV) Equation
4.8 Damped Forced MKdV (DFMKdV) Equation
4.9 Forced Schamel KdV (SKdV) Equation
4.10 Burgers’ Equation
4.11 Modified Burgers’ Equation
4.12 KdV Burgers’ (KdVB) Equation
4.13 Damped KdVB Equation
4.14 Kadomtsev–Petviashvili (KP) Equation
4.15 Modified KP (MKP) Equation
4.16 Further MKP (FMKP) Equation
4.17 KP Burgers’ (KPB) Equation
4.18 Damped KP (DKP) Equation
4.19 Zakharov–Kuznetsov (ZK) Equation
4.20 ZK Burgers’ (ZKB) Equation
4.21 Damped ZK (DZK) Equation
References
Chapter 5. Dressed Soliton and Envelope Soliton
5.1 Dressed Soliton
5.2 Dressed Soliton in a Classical Plasma
5.3 Dressed Soliton in a Dusty Plasma
5.4 Dressed Soliton in Quantum Plasma
5.5 Dressed Soliton of ZK Equation
5.6 Envelope Soliton
5.7 Nonlinear Schrodinger Equation (NLSE)
References
Chapter 6. Evolution Equations in Nonplanar Geometry
6.1 Introduction
6.2 Basic Equations of Motion in Nonplanar Geometry
6.3 Nonplanar KdV Equation in Classical Plasma
6.4 Nonplanar KdV Equation in Quantum Plasma
6.5 Nonplanar Gardner’s or Modified Gardner’s Equation
6.6 Nonplanar KP and KP Burgers’ Equation
6.7 Nonplanar ZK Equation
6.8 Nonplanar ZKB Equation
References
Chapter 7. Collision of Solitons
7.1 Introduction
7.2 Head-on Collision
7.2.1 Head-on collision of solitary waves in planar geometry
7.2.2 Head-on collision of solitons in a Magnetized Quantum Plasma
7.2.3 Head-on collision of magneto-acoustic solitons in spin-1/2 fermionic quantum plasma
7.2.4 Interaction of DIASWs in nonplanar geometry
7.3 Oblique Collision
7.3.1 Oblique collision of DIASWs in quantum plasmas
7.4 Overtaking Collision
7.4.1 Overtaking interaction of two solitons and three solitons of EAWs in quantum plasma
7.5 Soliton Interaction and Soliton Turbulence
7.6 Statistical Characteristics of the Wavefield
7.7 Plasma Parameters on Soliton Turbulence
References
Chapter 8. Sagdeev’s Pseudopotential Approach
8.1 Nonperturbative Approach
8.2 Sagdeev’s Pseudopotential Approach
8.2.1 Physical interpretation of Sagdeev’s potential
8.2.2 Determination of the range of Mach number
8.2.3 Shape of the solitary waves
8.2.4 Physical interpretation of double layers
8.2.5 Small amplitude approximation
8.3 Effect of Finite Ion Temperature
8.4 Large-amplitude DASWs
8.5 Large-amplitude Double Layers
8.6 Effect of Ion Kinematic Viscosity
8.7 DIASWs in Magnetized Plasma
8.8 Solitary Kinetic Alfven Waves
8.9 Collapse of EA Solitary Waves
8.10 Collapse of DASWs in Presence of Trapped Ions
References
Chapter 9. Conclusion and Future Scopes
Index
