Nonlinear Dynamics
Integrability, Chaos and Patterns
(Sprache: Englisch)
Covers all aspects of nonlinear dynamics in a unified and comprehensive way. Numerous examples and exercises will help the student to assimilate and apply the techniques presented. Suited to an interdisciplinary audience of physicists, engineers and other applied scientists.
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Produktinformationen zu „Nonlinear Dynamics “
Covers all aspects of nonlinear dynamics in a unified and comprehensive way. Numerous examples and exercises will help the student to assimilate and apply the techniques presented. Suited to an interdisciplinary audience of physicists, engineers and other applied scientists.
Klappentext zu „Nonlinear Dynamics “
This self-contained treatment covers all aspects of nonlinear dynamics, from fundamentals to recent developments, in a unified and comprehensive way. Numerous examples and exercises will help the student to assimilate and apply the techniques presented.
Inhaltsverzeichnis zu „Nonlinear Dynamics “
1. What is Nonlinearity?1.1 Dynamical Systems: Linear and Nonlinear Forces
1.2 Mathematical Implications of Nonlinearity
1.2.1 Linear and Nonlinear Systems
1.2.2 Linear Superposition Principle
1.3 Working Definition of Nonlinearity
1.4 Effects of Nonlinearity
2. Linear and Nonlinear Oscillators
2.1 Linear Oscillators and Predictability
2.1.1 Free Oscillations
2.1.2 Damped Oscillations
2.1.3 Damped and Forced Oscillations
2.2 Damped and Driven Nonlinear Oscillators
2.2.1 Free Oscillations
2.2.2 Damped Oscillations
2.2.3 Forced Oscillations - Primary Resonance and Jump Phenomenon (Hysteresis)
2.2.4 Secondary Resonances (Subharmonic and Superharmonic)
2.3 Nonlinear Oscillations and Bifurcations
- Problems
3. Qualitative Features
3.1 Autonomous and Nonautonomous Systems
3.2 Dynamical Systems as Coupled First-Order Differential Equations: Equilibrium Points
3.3 Phase Space/Phase Plane and Phase Trajectories: Stability, Attractors and Repellers
3.4 Classification of Equilibrium Points: Two-Dimensional Case
3.4.1 General Criteria for Stability
3.4.2 Classification of Equilibrium (Singular) Points
3.5 Limit Cycle Motion - Periodic Attractor
3.5.1 Poincaré-Bendixson Theorem
3.6 Higher Dimensional Systems
3.6.1 Example: Lorenz Equations
3.7 More Complicated Attractors
3.7.1 Torus
3.7.2 Quasiperiodic Attractor
3.7.3 Poincaré Map
3.7.4 Chaotic Attractor
3.8 Dissipative and Conservative Systems
3.8.1 Hamiltonian Systems
3.9 Conclusions
- Problems
4. Bifurcations and Onset of Chaos in Dissipative Systems
4.1 Some Simple Bifurcations
4.1.1 Saddle-Node Bifurcation
4.1.2 The Pitchfork Bifurcation
4.1.3 Transcritical Bifurcation
4.1.4 Hopf Bifurcation
4.2 Discrete Dynamical Systems
4.2.1 The Logistic Map
4.2.2 Equilibrium Points and Their Stability
4.2.3 Stability When the First Derivative Equals to +1 or -1
4.2.4 Periodic Solutions or Cycles
4.2.5 Period Doubling Phenomenon
4.2.6 Onset of Chaos: Sensitive Dependence on Initial
... mehr
Conditions - Lyapunov Exponent
4.2.7 Bifurcation Diagram
4.2.8 Bifurcation Structure in the Interval 3.57 ? a ? 4
4.2.9 Exact Solution at a = 4
4.2.10 Logistic Map: A Geometric Construction of the Dynamics - Cobweb Diagrams
4.3 Strange Attractor in the HŽenon Map
4.3.1 The Period Doubling Phenomenon
4.3.2 Self-Similar Structure
4.4 Other Routes to Chaos
4.4.1 Quasiperiodic Route to Chaos
4.4.2 Intermittency Route to Chaos
4.4.3 Type-I Intermittency
4.4.4 Standard Bifurcations in Maps
- Problems
5. Chaos in Dissipative Nonlinear Oscillators and Criteria for Chaos
5.1 Bifurcation Scenario in Duffing Oscillator
5.1.1 Period Doubling Route to Chaos
5.1.2 Intermittency Transition
5.1.3 Quasiperiodic Route to Chaos
5.1.4 Strange Nonchaotic Attractors (SNAs)
5.2 Lorenz Equations
5.2.1 Period Doubling Bifurcations and Chaos
5.3 Some Other Ubiquitous Chaotic Oscillators
5.3.1 Driven van der Pol Oscillator
5.3.2 Damped, Driven Pendulum
5.3.3 Morse Oscillator
5.3.4 Rössler Equations
5.4 Necessary Conditions for Occurrence of Chaos
5.4.1 Continuous Time Dynamical Systems (Differential Equations)
5.4.2 Discrete Time Systems (Maps)
5.5 Computational Chaos, Shadowing and All That
5.6 Conclusions
- Problems
6. Chaos in Nonlinear Electronic Circuits
6.1 Linear and Nonlinear Circuit Elements
6.2 Linear Circuits: The Resonant RLC Circuit
6.3 Nonlinear Circuits
6.3.1 Chua's Diode: Autonomous Case
6.3.2 A Simple Practical Implementation of Chua's Diode
6.3.3 Bifurcations and Chaos
6.4 Chaotic Dynamics of the Simplest Dissipative Nonautonomous Circuit: Murali-Lakshmanan-Chua (MLC) Circuit
6.4.1 Experimental Realization
6.4.2 Stability Analysis
6.4.3 Explicit Analytical Solutions
6.4.4 Experimental and Numerical Studies
6.5 Analog Circuit Simulations
6.6 Some Other Useful Nonlinear Circuits
6.6.1 RL Diode Circuit
6.6.2 Hunt's Nonlinear Oscillator
6.6.3 p-n Junction Diode Oscillator
6.6.4 Modified Chua Circuit
6.6.5 Colpitt's Oscillator
6.7 Nonlinear Circuits as Dynamical Systems
- Problems
7. Chaos in Conservative Systems
7.1 Poincaré Cross Section or Surface of Section
7.2 Possible Orbits in Conservative Systems
7.2.1 Regular Trajectories
7.2.2 Irregular Trajectories
7.2.3 Canonical Perturbation Theory: Overlapping Resonances and Chaos
7.3 Hénon-Heiles System
7.3.1 Equilibrium Points
7.3.2 Poincaré Surface of Section of the System
7.3.3 Numerical Results
7.4 Periodically Driven Undamped Duffing Oscillator
7.5 The Standard Map
7.5.1 Linear Stability and Invariant Curves
7.5.2 Numerical Analysis: Regular and Chaotic Motions
7.6 Kolmogorov-Arnold-Moser Theorem
7.7 Conclusions
- Problems
8. Characterization of Regular and Chaotic Motions
8.1 Lyapunov Exponents
8.2 Numerical Computation of Lyapunov Exponents
8.2.1 One-Dimensional Map
8.2.2 Computation of Lyapunov Exponents for Continuous Time Dynamical Systems
8.3 Power Spectrum
8.3.1 The Power Spectrum and Dynamical Motion
8.4 Autocorrelation
8.5 Dimension
8.6 Criteria for Chaotic Motion
- Problems
9. Further Developments in Chaotic Dynamics
9.1 Time Series Analysis
9.1.1 Estimation of Time-Delay and Embedding Dimension
9.1.2 Largest Lyapunov Exponent
- Problems
9.2 Stochastic Resonance
- Problems
9.3 Chaotic Scattering
- Problems
9.4 Controlling of Chaos
9.4.1 Controlling and Controlling Algorithms
9.4.2 Stabilization of UPO
- Problems
9.5 Synchronization of Chaos
9.5.1 Chaos in the DVP Oscillator
9.5.2 Synchronization of Chaos in the DVP Oscillator
9.5.3 Chaotic Signal Masking and Transmission of Analog Signals
9.5.4 Chaotic Digital Signal Transmission
- Problems
9.6 Quantum Chaos
9.6.1 Quantum Signatures of Chaos
9.6.2 Rydberg Atoms and Quantum Chaos
9.6.3 Hydrogen Atom in a Generalized van der Waals Interaction
9.6.4 Outlook
- Problems
9.7 Conclusions
10. Finite Dimensional Integrable Nonlinear Dynamical Systems
10.1 What is Integrability?
10.2 The Notion of Integrability
10.3 Complete Integrability - Complex Analytic Integrability
10.3.1 Real Time and Complex Time Behaviours
10.3.2 Partial Integrability and Constrained Integrability
10.3.3 Integrability and Separability
10.4 How to Detect Integrability: Painlevé Analysis
10.4.1 Classification of Singular Points
10.4.2 Historical Development of the Painlevé Approach and Integrability of Ordinary Differential Equations
10.4.3 Painlevé Method of Singular Point Analysis for Ordinary Differential Equations
10.5 Painlevé Analysis and Integrability of Two-Coupled Nonlinear Oscillators
10.5.1 Quartic Anharmonic Oscillators
10.6 Symmetries and Integrability
10.6.1 Invariance Conditions, Determination of Infinitesimals and First Integrals of Motion
10.6.2 Application - The Hénon-Heiles System
10.7 A Direct Method of Finding Integrals of Motion
10.8 Integrable Systems with Degrees of Freedom Greater Than Two
10.9 Integrable Discrete Systems
10.10 Integrable Dynamical Systems on Discrete Lattices
10.11 Conclusion
- Problems
11. Linear and Nonlinear Dispersive Waves
11.1 Linear Waves
11.2 Linear Nondispersive Wave Propagation
11.3 Linear Dispersive Wave Propagation
11.4 Fourier Transform and Solution of Initial Value Problem
11.5 Wave Packet and Dispersion
11.6 Nonlinear Dispersive Systems
11.6.1 An Illustration of the Wave of Permanence
11.6.2 John Scott Russel's Great Wave of Translation
11.7 Cnoidal and Solitary Waves
11.7.1 Korteweg-de Vries Equation and the Solitary Waves and Cnoidal Waves
11.8 Conclusions
- Problems
12. Korteweg-de Vries Equation and Solitons
12.1 The Scott Russel Phenomenon and KdV Equation
12.2 The Fermi-Pasta-Ulam Numerical Experiments on Anharmonic Lattices
12.2.1 The FPU Lattice
12.2.2 FPU Recurrence Phenomenon
12.3 The KdV Equation Again!
12.3.1 Asymptotic Analysis and the KdV Equation
12.4 Numerical Experiments of Zabusky and Kruskal: The Birth of Solitons
12.5 Hirota's Direct or Bilinearization Method for Soliton Solutions of KdV Equation
12.6 Conclusions
13. Basic Soliton Theory of KdV Equation
13.1 The Miura Transformation and Linearization of KdV: The Lax Pair
13.1.1 The Miura Transformation
13.1.2 Galilean Invariance and Schrödinger Eigenvalue Problem
13.1.3 Linearization of the KdV Equation
13.1.4 Lax Pair
13.2 Lax Pair and the Method of Inverse Scattering: A New Method to Solve the Initial Value Problem
13.2.1 The Inverse Scattering Transform (IST) Method for KdV Equation
13.3 Explicit Soliton Solutions
13.3.1 One-Soliton Solution (N = 1)
13.3.2 Two-Soliton Solution
13.3.3 N-Soliton Solution
13.3.4 Soliton Interaction
13.3.5 Nonreflectionless Potentials
13.4 Hamiltonian Structure of KdV Equation
13.4.1 Dynamics of Continuous Systems
13.4.2 KdV as a Hamiltonian Dynamical System
13.4.3 Complete Integrability of the KdV Equation
13.5 Infinite Number of Conserved Densities
13.6 Bäcklund Transformations
13.7 Conclusions
14. Other Ubiquitous Soliton Equations
14.1 Identification of Some Ubiquitous Nonlinear Evolution Equations from Physical Problems
14.1.1 The Nonlinear Schrödinger Equation in Optical Fibers
14.1.2 The Sine-Gordon Equation in Long Josephson Junctions
14.1.3 Dynamics of Ferromagnets:
Heisenberg Spin Equations
14.1.4 The Lattice with Exponential Interaction: The Toda Equation
14.2 The Zakharov-Shabat (ZS)/ Ablowitz-Kaup-Newell-Segur (AKNS) Linear Eigenvalue Problem and NLEES
14.2.1 The AKNS Linear Eigenvalue Problem and AKNS Equations
14.2.2 The Standard Soliton Equations
14.3 Solitary Wave Solutions and Basic Solitons
14.3.1 The MKdV Equation: Pulse Soliton
14.3.2 The sine-Gordon Equation: Kink, Antikink and Breathers
14.3.3 The Nonlinear Schršodinger Equation: Envelope Soliton
14.3.4 The Heisenberg Spin Equation: The Spin Soliton
14.3.5 The Toda Lattice: Discrete Soliton
14.4 Hirota's Method and Soliton Nature of Solitary Waves
14.4.1 The Modified KdV Equation
14.4.2 The NLS Equation
14.4.3 The sine-Gordon Equation
14.4.4 The Heisenberg Spin System
14.5 Solutions via IST Method
14.5.1 Direct and Inverse Scattering
14.5.2 Time Evolution of the Scattering Data
14.5.3 Soliton Solutions
14.6 Bäcklund Transformations
14.7 Conservation Laws and Constants of Motion
14.8 Hamiltonian Structure and Integrability
14.8.1 Hamiltonian Structure
14.8.2 Complete Integrability of the NLS Equation
14.9 Conclusions
- Problems
15. Spatio-Temporal Patterns
15.1 Linear Diffusion Equation
15.2 Nonlinear Diffusion and Reaction-Diffusion Equations
15.2.1 Nonlinear Reaction-Diffusion Equations
15.2.2 Dissipative Systems
15.3 Spatio-Temporal Patterns in Reaction-Diffusion Systems
15.3.1 Homogeneous Patterns
15.3.2 Autowaves: Travelling Wave Fronts, Pulses, etc
15.3.3 Ring Waves, Spiral Waves and Scroll Waves
15.3.4 Turing Instability and Turing Patterns
15.3.5 Localized Structures
15.3.6 Spatio-Temporal Chaos
15.4 Cellular Neural/Nonlinear Networks (CNNs)
15.4.1 Cellular Nonlinear Networks (CNNs)
15.4.2 Arrays of MLC Circuits: Simple Examples of CNN
15.4.3 Active Wave Propagation and its Failure in One-Dimensional CNNs
15.4.4 Turing Patterns
15.4.5 Spatio-Temporal Chaos
15.5 Some Exactly Solvable Nonlinear Diffusion Equations
15.5.1 The Burgers Equation
15.5.2 The Fokas-Yortsos-Rosen Equation
15.5.3 Generalized Fisher's Equation
15.6 Conclusion
- Problems
16. Nonlinear Dynamics: From Theory to Technology
16.1 Chaotic Cryptography
16.1.1 Basic Idea of Cryptography
16.1.2 An Elementary Chaotic Cryptographic System
16.2 Using Chaos (Controlling) to Calm the Web
16.3 Some Other Possibilities of Using Chaos
16.3.1 Communicating by Chaos
16.3.2 Chaos and Financial Markets
16.4 Optical Soliton Based Communications
16.5 Soliton Based Optical Computing
16.5.1 Photo-Refractive Materials and the Manakov Equation
16.5.2 Soliton Solutions and Shape Changing Collisions
16.5.3 Optical Soliton Based Computation
16.6 Micromagnetics and Magnetoelectronics
16.7 Conclusions
A. Elliptic Functions and Solutions of Certain Nonlinear Equations
- Problems
B. Perturbation and Related Approximation Methods
- B.1 Approximation Methods for Nonlinear Differential Equations
- B.2 Canonical Perturbation Theory for Conservative Systems
- B.2.1 One Degree ol Freedom Hamiltonian Systems
- B.2.2 Two Degrees ol Freedom Systems
- Problems
C. A Fourth-Order Runge-Kutta Integration Method
- Problems
- Problems
E. Fractals and Multifractals
- Problems
- Problems
G. Inverse Scattering Transform for the Schrödinger Spectral Problem
- G.l The Linear Eigenvalue Problem
- G.2 The Direct Scattering Problem
- G.3 The Inverse Scattering Problem
- G.4 Reconstruction of the Potential
- Problems
H. Inverse Scattering Transform for the Zakharov-Shabat Eigenvalue Problem
- H.1 The Linear Eigenvalue Problem
- H.2 The Direct Scattering Problem
- H.3 Inverse Scattering Problem
- H.4 Reconstruction of the Potentials
- Problems
I. Integrable Discrete Soliton Systems
I.1 Integrable Finite Dimensional N-Particles System on a Line: Calogero-Moser System
I.2 The Toda Lattice
I.3 Other Discrete Lattice Systems
I.4 Solitary Wave (Soliton) Solution of the Toda Lattice
- Problems
J. Painlevé Analysis for Partial Differential Equations
- J.1 The Painlevé Property for PDEs
- J.1.1 Painlevé Analysis
- J.2 Examples
- J.2.1 KdV Equation
- J.2.2 The Nonlinear Schrödinger Equation
- Problems
- References
4.2.7 Bifurcation Diagram
4.2.8 Bifurcation Structure in the Interval 3.57 ? a ? 4
4.2.9 Exact Solution at a = 4
4.2.10 Logistic Map: A Geometric Construction of the Dynamics - Cobweb Diagrams
4.3 Strange Attractor in the HŽenon Map
4.3.1 The Period Doubling Phenomenon
4.3.2 Self-Similar Structure
4.4 Other Routes to Chaos
4.4.1 Quasiperiodic Route to Chaos
4.4.2 Intermittency Route to Chaos
4.4.3 Type-I Intermittency
4.4.4 Standard Bifurcations in Maps
- Problems
5. Chaos in Dissipative Nonlinear Oscillators and Criteria for Chaos
5.1 Bifurcation Scenario in Duffing Oscillator
5.1.1 Period Doubling Route to Chaos
5.1.2 Intermittency Transition
5.1.3 Quasiperiodic Route to Chaos
5.1.4 Strange Nonchaotic Attractors (SNAs)
5.2 Lorenz Equations
5.2.1 Period Doubling Bifurcations and Chaos
5.3 Some Other Ubiquitous Chaotic Oscillators
5.3.1 Driven van der Pol Oscillator
5.3.2 Damped, Driven Pendulum
5.3.3 Morse Oscillator
5.3.4 Rössler Equations
5.4 Necessary Conditions for Occurrence of Chaos
5.4.1 Continuous Time Dynamical Systems (Differential Equations)
5.4.2 Discrete Time Systems (Maps)
5.5 Computational Chaos, Shadowing and All That
5.6 Conclusions
- Problems
6. Chaos in Nonlinear Electronic Circuits
6.1 Linear and Nonlinear Circuit Elements
6.2 Linear Circuits: The Resonant RLC Circuit
6.3 Nonlinear Circuits
6.3.1 Chua's Diode: Autonomous Case
6.3.2 A Simple Practical Implementation of Chua's Diode
6.3.3 Bifurcations and Chaos
6.4 Chaotic Dynamics of the Simplest Dissipative Nonautonomous Circuit: Murali-Lakshmanan-Chua (MLC) Circuit
6.4.1 Experimental Realization
6.4.2 Stability Analysis
6.4.3 Explicit Analytical Solutions
6.4.4 Experimental and Numerical Studies
6.5 Analog Circuit Simulations
6.6 Some Other Useful Nonlinear Circuits
6.6.1 RL Diode Circuit
6.6.2 Hunt's Nonlinear Oscillator
6.6.3 p-n Junction Diode Oscillator
6.6.4 Modified Chua Circuit
6.6.5 Colpitt's Oscillator
6.7 Nonlinear Circuits as Dynamical Systems
- Problems
7. Chaos in Conservative Systems
7.1 Poincaré Cross Section or Surface of Section
7.2 Possible Orbits in Conservative Systems
7.2.1 Regular Trajectories
7.2.2 Irregular Trajectories
7.2.3 Canonical Perturbation Theory: Overlapping Resonances and Chaos
7.3 Hénon-Heiles System
7.3.1 Equilibrium Points
7.3.2 Poincaré Surface of Section of the System
7.3.3 Numerical Results
7.4 Periodically Driven Undamped Duffing Oscillator
7.5 The Standard Map
7.5.1 Linear Stability and Invariant Curves
7.5.2 Numerical Analysis: Regular and Chaotic Motions
7.6 Kolmogorov-Arnold-Moser Theorem
7.7 Conclusions
- Problems
8. Characterization of Regular and Chaotic Motions
8.1 Lyapunov Exponents
8.2 Numerical Computation of Lyapunov Exponents
8.2.1 One-Dimensional Map
8.2.2 Computation of Lyapunov Exponents for Continuous Time Dynamical Systems
8.3 Power Spectrum
8.3.1 The Power Spectrum and Dynamical Motion
8.4 Autocorrelation
8.5 Dimension
8.6 Criteria for Chaotic Motion
- Problems
9. Further Developments in Chaotic Dynamics
9.1 Time Series Analysis
9.1.1 Estimation of Time-Delay and Embedding Dimension
9.1.2 Largest Lyapunov Exponent
- Problems
9.2 Stochastic Resonance
- Problems
9.3 Chaotic Scattering
- Problems
9.4 Controlling of Chaos
9.4.1 Controlling and Controlling Algorithms
9.4.2 Stabilization of UPO
- Problems
9.5 Synchronization of Chaos
9.5.1 Chaos in the DVP Oscillator
9.5.2 Synchronization of Chaos in the DVP Oscillator
9.5.3 Chaotic Signal Masking and Transmission of Analog Signals
9.5.4 Chaotic Digital Signal Transmission
- Problems
9.6 Quantum Chaos
9.6.1 Quantum Signatures of Chaos
9.6.2 Rydberg Atoms and Quantum Chaos
9.6.3 Hydrogen Atom in a Generalized van der Waals Interaction
9.6.4 Outlook
- Problems
9.7 Conclusions
10. Finite Dimensional Integrable Nonlinear Dynamical Systems
10.1 What is Integrability?
10.2 The Notion of Integrability
10.3 Complete Integrability - Complex Analytic Integrability
10.3.1 Real Time and Complex Time Behaviours
10.3.2 Partial Integrability and Constrained Integrability
10.3.3 Integrability and Separability
10.4 How to Detect Integrability: Painlevé Analysis
10.4.1 Classification of Singular Points
10.4.2 Historical Development of the Painlevé Approach and Integrability of Ordinary Differential Equations
10.4.3 Painlevé Method of Singular Point Analysis for Ordinary Differential Equations
10.5 Painlevé Analysis and Integrability of Two-Coupled Nonlinear Oscillators
10.5.1 Quartic Anharmonic Oscillators
10.6 Symmetries and Integrability
10.6.1 Invariance Conditions, Determination of Infinitesimals and First Integrals of Motion
10.6.2 Application - The Hénon-Heiles System
10.7 A Direct Method of Finding Integrals of Motion
10.8 Integrable Systems with Degrees of Freedom Greater Than Two
10.9 Integrable Discrete Systems
10.10 Integrable Dynamical Systems on Discrete Lattices
10.11 Conclusion
- Problems
11. Linear and Nonlinear Dispersive Waves
11.1 Linear Waves
11.2 Linear Nondispersive Wave Propagation
11.3 Linear Dispersive Wave Propagation
11.4 Fourier Transform and Solution of Initial Value Problem
11.5 Wave Packet and Dispersion
11.6 Nonlinear Dispersive Systems
11.6.1 An Illustration of the Wave of Permanence
11.6.2 John Scott Russel's Great Wave of Translation
11.7 Cnoidal and Solitary Waves
11.7.1 Korteweg-de Vries Equation and the Solitary Waves and Cnoidal Waves
11.8 Conclusions
- Problems
12. Korteweg-de Vries Equation and Solitons
12.1 The Scott Russel Phenomenon and KdV Equation
12.2 The Fermi-Pasta-Ulam Numerical Experiments on Anharmonic Lattices
12.2.1 The FPU Lattice
12.2.2 FPU Recurrence Phenomenon
12.3 The KdV Equation Again!
12.3.1 Asymptotic Analysis and the KdV Equation
12.4 Numerical Experiments of Zabusky and Kruskal: The Birth of Solitons
12.5 Hirota's Direct or Bilinearization Method for Soliton Solutions of KdV Equation
12.6 Conclusions
13. Basic Soliton Theory of KdV Equation
13.1 The Miura Transformation and Linearization of KdV: The Lax Pair
13.1.1 The Miura Transformation
13.1.2 Galilean Invariance and Schrödinger Eigenvalue Problem
13.1.3 Linearization of the KdV Equation
13.1.4 Lax Pair
13.2 Lax Pair and the Method of Inverse Scattering: A New Method to Solve the Initial Value Problem
13.2.1 The Inverse Scattering Transform (IST) Method for KdV Equation
13.3 Explicit Soliton Solutions
13.3.1 One-Soliton Solution (N = 1)
13.3.2 Two-Soliton Solution
13.3.3 N-Soliton Solution
13.3.4 Soliton Interaction
13.3.5 Nonreflectionless Potentials
13.4 Hamiltonian Structure of KdV Equation
13.4.1 Dynamics of Continuous Systems
13.4.2 KdV as a Hamiltonian Dynamical System
13.4.3 Complete Integrability of the KdV Equation
13.5 Infinite Number of Conserved Densities
13.6 Bäcklund Transformations
13.7 Conclusions
14. Other Ubiquitous Soliton Equations
14.1 Identification of Some Ubiquitous Nonlinear Evolution Equations from Physical Problems
14.1.1 The Nonlinear Schrödinger Equation in Optical Fibers
14.1.2 The Sine-Gordon Equation in Long Josephson Junctions
14.1.3 Dynamics of Ferromagnets:
Heisenberg Spin Equations
14.1.4 The Lattice with Exponential Interaction: The Toda Equation
14.2 The Zakharov-Shabat (ZS)/ Ablowitz-Kaup-Newell-Segur (AKNS) Linear Eigenvalue Problem and NLEES
14.2.1 The AKNS Linear Eigenvalue Problem and AKNS Equations
14.2.2 The Standard Soliton Equations
14.3 Solitary Wave Solutions and Basic Solitons
14.3.1 The MKdV Equation: Pulse Soliton
14.3.2 The sine-Gordon Equation: Kink, Antikink and Breathers
14.3.3 The Nonlinear Schršodinger Equation: Envelope Soliton
14.3.4 The Heisenberg Spin Equation: The Spin Soliton
14.3.5 The Toda Lattice: Discrete Soliton
14.4 Hirota's Method and Soliton Nature of Solitary Waves
14.4.1 The Modified KdV Equation
14.4.2 The NLS Equation
14.4.3 The sine-Gordon Equation
14.4.4 The Heisenberg Spin System
14.5 Solutions via IST Method
14.5.1 Direct and Inverse Scattering
14.5.2 Time Evolution of the Scattering Data
14.5.3 Soliton Solutions
14.6 Bäcklund Transformations
14.7 Conservation Laws and Constants of Motion
14.8 Hamiltonian Structure and Integrability
14.8.1 Hamiltonian Structure
14.8.2 Complete Integrability of the NLS Equation
14.9 Conclusions
- Problems
15. Spatio-Temporal Patterns
15.1 Linear Diffusion Equation
15.2 Nonlinear Diffusion and Reaction-Diffusion Equations
15.2.1 Nonlinear Reaction-Diffusion Equations
15.2.2 Dissipative Systems
15.3 Spatio-Temporal Patterns in Reaction-Diffusion Systems
15.3.1 Homogeneous Patterns
15.3.2 Autowaves: Travelling Wave Fronts, Pulses, etc
15.3.3 Ring Waves, Spiral Waves and Scroll Waves
15.3.4 Turing Instability and Turing Patterns
15.3.5 Localized Structures
15.3.6 Spatio-Temporal Chaos
15.4 Cellular Neural/Nonlinear Networks (CNNs)
15.4.1 Cellular Nonlinear Networks (CNNs)
15.4.2 Arrays of MLC Circuits: Simple Examples of CNN
15.4.3 Active Wave Propagation and its Failure in One-Dimensional CNNs
15.4.4 Turing Patterns
15.4.5 Spatio-Temporal Chaos
15.5 Some Exactly Solvable Nonlinear Diffusion Equations
15.5.1 The Burgers Equation
15.5.2 The Fokas-Yortsos-Rosen Equation
15.5.3 Generalized Fisher's Equation
15.6 Conclusion
- Problems
16. Nonlinear Dynamics: From Theory to Technology
16.1 Chaotic Cryptography
16.1.1 Basic Idea of Cryptography
16.1.2 An Elementary Chaotic Cryptographic System
16.2 Using Chaos (Controlling) to Calm the Web
16.3 Some Other Possibilities of Using Chaos
16.3.1 Communicating by Chaos
16.3.2 Chaos and Financial Markets
16.4 Optical Soliton Based Communications
16.5 Soliton Based Optical Computing
16.5.1 Photo-Refractive Materials and the Manakov Equation
16.5.2 Soliton Solutions and Shape Changing Collisions
16.5.3 Optical Soliton Based Computation
16.6 Micromagnetics and Magnetoelectronics
16.7 Conclusions
A. Elliptic Functions and Solutions of Certain Nonlinear Equations
- Problems
B. Perturbation and Related Approximation Methods
- B.1 Approximation Methods for Nonlinear Differential Equations
- B.2 Canonical Perturbation Theory for Conservative Systems
- B.2.1 One Degree ol Freedom Hamiltonian Systems
- B.2.2 Two Degrees ol Freedom Systems
- Problems
C. A Fourth-Order Runge-Kutta Integration Method
- Problems
- Problems
E. Fractals and Multifractals
- Problems
- Problems
G. Inverse Scattering Transform for the Schrödinger Spectral Problem
- G.l The Linear Eigenvalue Problem
- G.2 The Direct Scattering Problem
- G.3 The Inverse Scattering Problem
- G.4 Reconstruction of the Potential
- Problems
H. Inverse Scattering Transform for the Zakharov-Shabat Eigenvalue Problem
- H.1 The Linear Eigenvalue Problem
- H.2 The Direct Scattering Problem
- H.3 Inverse Scattering Problem
- H.4 Reconstruction of the Potentials
- Problems
I. Integrable Discrete Soliton Systems
I.1 Integrable Finite Dimensional N-Particles System on a Line: Calogero-Moser System
I.2 The Toda Lattice
I.3 Other Discrete Lattice Systems
I.4 Solitary Wave (Soliton) Solution of the Toda Lattice
- Problems
J. Painlevé Analysis for Partial Differential Equations
- J.1 The Painlevé Property for PDEs
- J.1.1 Painlevé Analysis
- J.2 Examples
- J.2.1 KdV Equation
- J.2.2 The Nonlinear Schrödinger Equation
- Problems
- References
... weniger
Bibliographische Angaben
- Autoren: Muthusamy Lakshmanan , Shanmuganathan Rajaseekar
- 2003, Repr. d. Ausg. v. 2002., 620 Seiten, Masse: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 3540439080
- ISBN-13: 9783540439080
- Erscheinungsdatum: 12.11.2002
Sprache:
Englisch
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