Dynamical Systems and Nonlinear Waves in Plasmas

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. This book provides clear ideas on dynamical systems and waves in plasmas. It also presents derivation of different nonlinear evolution equations describing nonlinear plasma waves (ion-acoustic, dust-acoustic, dust-ion-acoustic, electron-acoustic) through a perturbative approach. Cover 1 Title Page 2 Copyright Page 3 Dedication 4 Preface 5 Table of Contents 7 1. Introduction 12 1.1 Plasma as a state 12 1.2 Plasmas exist in nature 13 1.2.1 Ionosphere 13 1.2.2 Van Allen belts 13 1.2.3 Aurorae 14 1.2.4 Solar corona 14 1.2.5 Core of the sun 14 1.2.6 HII regions 15 1.3 Concept of temperature 15 1.3.1 Existence of several temperatures 18 1.3.2 Electron and ion temperatures 19 1.3.3 Quasineutrality in plasma 19 1.4 Debye length and Debye sphere 20 1.5 Criteria for plasma 23 1.6 Plasma frequency 24 1.7 Applications of plasma 26 1.7.1 Space physics 26 1.7.2 Astrophysics 26 1.7.3 Gas lasers 27 1.7.4 Industrial application 27 1.8 Fluid description of plasma 27 1.8.1 Maxwell’s equation 28 1.8.2 Equation of motion 28 1.8.3 Equation of continuity 30 1.8.4 Equation of state of plasma 31 1.8.5 Poisson equation 32 References 33 2. Dynamical Systems 34 2.1 Introduction to dynamical systems 34 2.1.1 One-dimensional system 34 2.1.1.1 Equilibrium point and its stability 35 2.1.1.2 Trajectory and phase portrait 35 2.1.1.3 Example 35 2.1.2 Linear stability analysis 35 2.1.2.1 Example 37 2.1.3 Potentials 37 2.1.3.1 Example 38 2.1.3.2 Example 38 2.1.4 Bifurcations 39 2.1.5 Linear system in two-dimension 40 2.1.5.1 Example 42 2.1.6 Phase plane analysis 43 2.1.6.1 Nonlinear system in two-dimension 44 2.1.6.2 Conservative system 45 2.1.6.3 Example 45 2.1.6.4 Example 47 2.1.6.5 Hamiltonian system 48 2.1.6.6 Example 48 References 49 3. Waves in Plasmas 50 3.1 Introduction to wave modes 50 3.1.1 Ion-acoustic (IA) waves 50 3.1.2 Dust-acoustic (DA) waves 51 3.1.3 Dust-ion-acoustic (DIA) waves 52 3.1.4 Upper hybrid wave 53 3.1.5 Electrostatic cyclotron waves 57 3.1.6 Lower hybrid wave 61 3.2 Reductive perturbation technique and evolution equations 65 3.2.1 The KdV equation 67 3.2.2 The Burgers equation 71 3.2.3 The KP equation 74 3.2.4 The ZK and mZK equations 78 3.3 Analytical wave solutions of evolution equations 82 3.3.1 Analytical wave solution of the KdV equation 82 3.3.2 Analytical wave solution of the mKdV equation 84 3.3.3 Analytical wave solution of the KP equation 86 3.3.4 Analytical wave solution of the mKP equation 88 3.3.5 Analytical wave solution of the ZK equation 91 3.3.6 Analytical wave solution of the mZK equation 93 3.3.7 Analytical wave solution of the Burgers equation 95 References 98 4. Bifurcation of Small Amplitude Waves in Plasmas 99 4.1 Introduction 99 4.2 Bifurcation of ion-acoustic waves with small amplitude 100 4.2.1 Basic equations 100 4.2.2 Derivation of the KdV equation 101 4.2.3 Formation of dynamical system 102 4.2.4 Phase plane analysis 102 4.2.5 Wave solutions 104 4.3 Bifurcation of dust-ion-acoustic waves with small amplitude 107 4.3.1 Governing equations 107 4.3.2 Derivation of the KP equation 109 4.3.3 Formation of dynamical system and phase portraits 110 4.3.4 Wave solutions 113 4.4 Bifurcation of dust-acoustic waves with small amplitude 116 4.4.1 Basic equations 117 4.4.2 Derivation of the Burgers equation 118 4.4.3 Formation of dynamical system and phase portraits 119 4.4.4 Wave solutions 120 4.5 Bifurcation of electron-acoustic waves with small amplitude 124 4.5.1 Basic equations 125 4.5.2 Derivation of the KdV equation 125 4.5.3 Formation of dynamical system and phase portraits 127 4.5.4 Wave solutions 128 References 132 5. Bifurcation of Arbitrary Amplitude Waves in Plasmas 137 5.1 Introduction 137 5.2 Bifurcation of ion-acoustic waves with arbitrary amplitude 138 5.2.1 Basic equations 138 5.2.2 Formation of dynamical system and phase portraits 138 5.2.3 Wave solutions 142 5.3 Bifurcation of dust-ion-acoustic waves with arbitrary amplitude 143 5.3.1 Basic equations 143 5.3.2 Formation of dynamical system and phase portraits 145 5.3.3 Wave solutions 150 5.4 Bifurcation of dust-acoustic waves with arbitrary amplitude 151 5.4.1 Basic equations 151 5.4.2 Formation of dynamical system and phase portraits 151 5.4.3 Wave solutions 154 5.5 Bifurcation of electron-acoustic waves with arbitrary amplitude 154 5.5.1 Basic equations 157 5.5.2 Formation of dynamical system and phase portraits 157 5.5.3 Wave solutions 160 References 163 6. Bifurcation Analysis of Supernonlinear Waves 165 6.1 Introduction: supernonlinear waves 165 6.1.1 Different kind of trajectories 165 6.2 Bifurcation of supernonlinear ion-acoustic waves 166 6.2.1 Basic equations 166 6.2.2 Modified KdV equation 169 6.2.3 Formation of dynamical system and phase portraits 170 6.2.4 Wave solutions 172 6.3 Bifurcation of supernonlinear dust-acoustic waves 173 6.3.1 Basic equations 173 6.3.2 Formation of dynamical system and phase portraits 174 6.3.3 Wave solutions 175 6.4 Bifurcation of supernonlinear electron-acoustic waves (EAWs) 180 6.4.1 Basic equations 181 6.4.2 The evolution equation and dynamical system 182 6.4.3 Wave solutions 184 References 189 7. Chaos, Multistability and Stable Oscillation in Plasmas 191 7.1 Chaos in a conservative dusty plasma 191 7.1.1 Basic equations 193 7.1.2 Multiperiodic, quasiperiodic and chaotic oscillations 193 7.2 Multistability of electron-acoustic waves 198 7.2.1 Basic equations 198 7.2.2 Multistability 199 7.3 Stable oscillation in a dissipative plasma 202 7.3.1 Model equations 203 7.3.2 The KdV-Burgers equation 204 7.3.3 Stability analysis of DAWs 206 References 211 Index 216 Phase,portrait;,Reductive,perturbation,technique;,Hamiltonian,system,and,function;,Chaos;,Hyperchaos;,Multistability;,Quasiperiodicity;,Supersolitary,wave,or,supersoliton;,Superperiodic,wave;,Supernonlinear,wave;,Conservative,system;,Analytical,wave,solution;,Nonlinear,evolution,equations Phase portrait,Reductive perturbation technique,Hamiltonian system and function,Chaos,Hyperchaos,Multistability,Quasiperiodicity,Supersolitary wave or supersoliton,Superperiodic wave,Supernonlinear wave,Conservative system,Analytical wave solution,Nonlinear evolution equations "This book provides clear ideas on dynamical systems and waves in plasmas. It also presents derivation of different nonlinear evolution equations describing nonlinear plasma waves (ion-acoustic, dust-acoustic, dust-ion-acoustic, electron-acoustic) through a perturbative approach. It demonstrates bifurcation behavior of nonlinear and supernonlinear waves characterized by trivial and nontrivial topologies of their phase portraits for the corresponding dynamical systems. Chaotic and hyperchaotic behaviors of various plasma waves are presented in presence or absence of external periodic force. It also provides real applications to NLSE and laser plasma interaction"-- Provided by publisher