Narrow Results

Using the Darboux transformation method, the general Lax equation is solved and a collection of new exact solutions together with one-soliton solutions, singular one-soliton solutions, periodic solutions, singular periodic solution, two-soliton solutions, singular two-soliton solutions, two-periodic solutions and singular two-periodic solutions is obtained. Using traveling wave transformation, the Lax equation is transfigured to a conservative dynamical system (CDS) of dimension four with three equilibrium points involving two parameters $\gamma$ and v. The CDS has various quasi-periodic motions for fixed values of the parameters $\gamma$ and v at different initial conditions. Furthermore, effects of the parameters $\gamma$ and v are shown on the quasiperiodic motions of the CDS by means of phase sections and time series plots. This approach can be applied to a heterogeneity of nonlinear model equations or partial differential equations for describing their inherent nonlinear phenomena.

The following sections are included:

• Introduction
• Saha Equation and Plasma Temperature
• Basic Concepts of Plasma
  • Basic dimensionless parameters
  • Debye length and Debye shielding
  • Quasineutrality
  • Response time
  • Plasma frequency
  • Collisions and coupling limit
• Criteria for Plasma
• High-Temperature Plasmas
• Mathematical Description
• Magnetized Plasmas
• Single Particle Motion in Uniform Electric and Magnetic Field
• Fluid Approach
• Maxwell’s Equations
• Electromagnetic Wave Equation in Free Space
• Plasma Kinetic Theory
  • Distribution function
  • Macroscopic variables
  • Maxwellian distribution function
  • Non-Maxwellian distribution in plasmas
  • Nonthermal distribution
  • Superthermal distribution
  • q-nonextensive distribution
• Closure Form of Moment Equation
  • Equation of continuity
  • Equation of motion
  • Equation of energy
• Dusty Plasma
• Quantum Plasma
• Quantum Plasma Models
• References

The following sections are included:

• Introduction
• Mathematical Description of Waves
• Dispersion Relation
• Linear Waves in Plasmas
• Plasma Oscillation
• Electromagnetic Waves
• Upper Hybrid Frequency
• Electrostatic Ion Cyclotron Waves
• Lower Hybrid Frequency
• Electromagnetic Waves with ${\overrightarrow{B}}_0=0$
• Electromagnetic Waves Perpendicular to B₀
  • Ordinary wave
  • Extraordinary wave
• Electromagnetic Waves Parallel to ${\overrightarrow{B}}_0$
• Hydromagnetic Waves
  • Alfven wave
  • Magnetosonic wave
• Some Acoustic Type of Waves in Plasmas
  • Electron plasma waves
  • Ion acoustic waves
  • Dust acoustic waves
  • Dust ion acoustic waves
• Nonlinear Wave
• Solitary Waves and Solitons
  • History of solitary waves and solitons
• Properties of Solitons
• References

The following sections are included:

• Nonlinear Waves
• Direct Method
  • Korteweg–de Vries (KdV) equation
  • Cnoidal waves
  • Modified KdV (MKdV) equation
  • Schamel-type KdV (S-KdV) equation
  • Burgers’ equation
  • KP equation
  • Modified KP equation
• Hyperbolic Tangent Method
  • KdV equation
  • Modified KdV equation
  • Burgers’ equation
  • KdV Burgers’ equation
  • KP equation
• Tanh–Coth Method
  • KdV equation
  • Burgers’ equation
• Solution of KP Burger Equation
• Conservation Laws and Integrals of the Motions
  • Conserved quantity of KdV equation
• Approximate Analytical Solutions
  • Damped KdV equation
  • Force KdV equation
  • Damped-force KdV equation
• Multisoliton and Hirota’s Direct Method
  • Hirota’s method
  • Multisoliton solution of the KdV equation
  • Multisoliton solution of the KP equation
• References

The following sections are included:

• Perturbation Technique
• Reductive Perturbation Technique
• Korteweg–de Vries (KdV) Equation
• Modified KdV (MKdV) Equation
• Gardner’s Equation
• Gardner and Modified Gardner’s (MG) Equation
• Damped Forced KdV (DFKdV) Equation
• Damped Forced MKdV (DFMKdV) Equation
• Forced Schamel KdV (SKdV) Equation
• Burgers’ Equation
• Modified Burgers’ Equation
• KdV Burgers’ (KdVB) Equation
• Damped KdVB Equation
• Kadomtsev–Petviashvili (KP) Equation
• Modified KP (MKP) Equation
• Further MKP (FMKP) Equation
• KP Burgers’ (KPB) Equation
• Damped KP (DKP) Equation
• Zakharov–Kuznetsov (ZK) Equation
• ZK Burgers’ (ZKB) Equation
• Damped ZK (DZK) Equation
• References

The following sections are included:

• Dressed Soliton
• Dressed Soliton in a Classical Plasma
• Dressed Soliton in a Dusty Plasma
• Dressed Soliton in Quantum Plasma
• Dressed Soliton of ZK Equation
• Envelope Soliton
• Nonlinear Schrodinger Equation (NLSE)
• References

The following sections are included:

• Introduction
• Basic Equations of Motion in Nonplanar Geometry
• Nonplanar KdV Equation in Classical Plasma
• Nonplanar KdV Equation in Quantum Plasma
• Nonplanar Gardner’s or Modified Gardner’s Equation
• Nonplanar KP and KP Burgers’ Equation
• Nonplanar ZK Equation
• Nonplanar ZKB Equation
• References

The following sections are included:

• Introduction
• Head-on Collision
  • Head-on collision of solitary waves in planar geometry
  • Head-on collision of solitons in a Magnetized Quantum Plasma
  • Head-on collision of magneto-acoustic solitons in spin-1/2 fermionic quantum plasma
  • Interaction of DIASWs in nonplanar geometry
• Oblique Collision
  • Oblique collision of DIASWs in quantum plasmas
• Overtaking Collision
  • Overtaking interaction of two solitons and three solitons of EAWs in quantum plasma
• Soliton Interaction and Soliton Turbulence
• Statistical Characteristics of the Wavefield
• Plasma Parameters on Soliton Turbulence
• References

The following sections are included:

• Nonperturbative Approach
• Sagdeev’s Pseudopotential Approach
  • Physical interpretation of Sagdeev’s potential
  • Determination of the range of Mach number
  • Shape of the solitary waves
  • Physical interpretation of double layers
  • Small amplitude approximation
• Effect of Finite Ion Temperature
• Large-amplitude DASWs
• Large-amplitude Double Layers
• Effect of Ion Kinematic Viscosity
• DIASWs in Magnetized Plasma
• Solitary Kinetic Alfven Waves
• Collapse of EA Solitary Waves
• Collapse of DASWs in Presence of Trapped Ions
• References

In this book, we have studied the linear and nonlinear waves in different plasma environments (classical plasma, dusty plasma, and quantum plasma). Moreover, the interaction of waves in various plasma models is also investigated. We have considered the fluid approach for this study. To study linear waves, the fluid equations are linearized in the neighborhood of an equilibrium point, and the first-order perturbed quantities are considered wavy (proportional to e^kx–iωt, where k is the wavenumber and w is the natural frequency). The relationship between the frequency and the wave number is obtained and is called the dispersion relation. This relation gives different wave modes in plasma. Several wave modes and their features are investigated in various unmagnetized and magnetized plasmas…

The following section is included:

• Index

The following sections are included:

• Preface
• Acknowledgments
• Contents