Solitons

Series Preface
Volume Preface
Acknowledgment
Contents
About the Editor-in-Chief
About the Volume Editor
Contributors
Nonlinear Water Waves and Nonlinear Evolution Equations with Applications
Introduction
Variational Principles and the Euler-Lagrange Equations
The Euler Equation of Motion and Water Wave Problems
The Variational Principle for Nonlinear Water Waves
Basic Equations of Nonlinear Water Waves
Surface Waves on a Running Stream in Water of Arbitrary, But Uniform, Depth
Critical Values and Resonance-Type Effect
Nonlinear Theory of Water Waves by a Moving Pressure Distribution at Resonant Conditions
The Nonlinear Schrödinger Equation and Evolution of Wave Packets
Higher-Order Nonlinear Schrödinger Equations
The Davey-Stewartson (DS) Equations in Water of Finite Depth
The Camassa-Holm (CH) and Degasperis-Procesi (DP) Nonlinear Model Equations
Periodic and Solitary Waves with Constant Vorticity
Bibliography
Inverse Scattering Transform and the Theory of Solitons
Glossary
Definition of the Subject
Introduction
Inverse Scattering Transform
The Lax Method
The AKNS Method
Direct Scattering Problem
Time Evolution of the Scattering Data
Inverse Scattering Problem
Solitons
Future Directions
Bibliography
Primary Literature
Books and Reviews
Different Analytical Methods for Solving the Korteweg-de Vries Equation (KdV)
Glossary
Definition of the Subject
Introduction
The Generalized Hyperbolic Function-Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation
The Definition and Properties of Generalized Hyperbolic Functions
A New Higher Order and Higher Dimension Bäcklund Transformation Method to Construct an Auto-Bäcklund Transformation of the (2 ...
The Generalized Hyperbolic Function-Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Case 10
Case 11
Case 12
Case 13
Case 14
Case 15
Case 16
Case 17
Case 18
Case 19
Case 20
Case 21
Case 22
Case 23
Case 24
Case 25
The Generalized F-Expansion Method and Its Application in Another(2 + 1)-Dimensional KdV Equation
Summary of the Generalized F-Expansion Method
The Generalized F-Expansion Method to Find the Exact Solutions of the (2 + 1)-Dimensional KdV Equation
Case 1
Case 2
Case 3
Case 4
The Generalized Algebra Method and Its Application in (1 + 1)-Dimensional Generalized Variable - Coefficient KdV Equation
A New Transformation and a New Theorem
A New Mechanization Method to Find the Exact Solutions of a First-Order Nonlinear Ordinary Differential Equation with any Degr...
Summary of the Generalized Algebra Method
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
Step 7
The Generalized Algebra Method to Find New Non-traveling Waves Solutionsof the (1 + 1)-Dimensional Generalized Variable-Coeffi...
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Case 10
Case 11
Type 1
Type 2
Type 3
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
A New Exp-N Solitary-Like Method and Its Application in the (1 + 1)-Dimensional Generalized KdV Equation
Summary of the Exp-N Solitary-Like Method
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
The Application of the Exp-N Solitary-Like Method in the (1 + 1)-Dimensional Generalized KdV Equation
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Case 10
Case 11
Case 12
Case 13
Case 14
Case 15
Case 16
Case 17
Case 18
Case 19
The Exp-Bäcklund Transformation Method and Its Application in (1 + 1)-Dimensional KdV Equation
Summary of the Exp-Bäcklund Transformation Method
Step 1
Step 2
Step 3
Step 4
Step 5
Step 6
The Application of the Exp-Bäcklund Transformation Method in (1 + 1)-Dimensional KdV Equation
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Case 10
Case 11
Case 12
Future Directions
Acknowledgments
Bibliography
Primary Literature
Books and Reviews
History, Exact N-Soliton Solutions and Further Properties of the Korteweg-de Vries Equation (KdV)
Glossary
Definition of the Subject
Introduction
Inverse Scattering Transform for the KdV Equation
Exact N-soliton Solutions of the KdV Equation
Further Properties of the KdV Equation
Conservation Laws
The Lax Hierarchy
Future Directions
Bibliography
Primary Literature
Books and Reviews
Semi-analytical Methods for Solving the KdV and mKdV Equations
Glossary
Definition of the Subject
Introduction
An Analysis of the Semi-Analytical Methods and their Applications
Adomian Decomposition Method
Homotopy Analysis Method
Homotopy Perturbation Method
Variational Iteration Method
Numerical Experiments
Future Directions
Bibliography
Primary Literature
Books and Reviews
Some Numerical Methods for Solving the Korteweg-de Vries Equation (KdV)
Glossary
Definition of the Subject
Introduction
Some Numerical Methods for Solving the Korteweg-de Vries (KdV) Equation
The Adomian Decomposition Method (ADM)
The Homotopy Analysis Method(HAM)
The Variational Iteration Method(VIM)
The Homotopy Perturbation Method(HPM)
Numerical Applications and Comparisons
The ADM for Eq. (14)
The HAM for Eq. (34)
The VIM for Eq. (34)
The HPM for Eq. (34)
The EFDM for Eq. (34)
Conclusions and Discussions
Future Directions
Bibliography
Primary Literature
Books and Reviews
Nonlinear Internal Waves
Glossary
Introduction
Problem and Frame of Reference
Notation
Superscripts
Subscripts
Equations of Motion
The Shallow Water Theory
Verification of the Homogeneous Equations
Complete Determination of the Solution
Free Surface and Interface Elevations of Different Modes
Secular Term
Multiple-Scale Transformation of Variables
Derivation of the KdV Equation
Conclusion
Future Directions
Bibliography
Primary Literature
Books and Reviews
Partial Differential Equations that Lead to Solitons
Definition of the Subject
Introduction
Some Nonlinear Models that Lead to Solitons
Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
Example 9
Example 10
Example 11
Example 12
Example 13
Example 14
Example 15
Example 16
Future Directions
Bibliography
Primary Literature
Books and Reviews
Shallow Water Waves and Solitary Waves
Glossary
Definition of the Subject
Introduction
Completely Integrable Shallow Water Wave Equations
The Korteweg-de Vries Equation
Regularized Long-Wave Equations
The Boussinesq Equation
1D Shallow Water Wave Equation
The Camassa-Holm Equation
The Kadomtsev-Petviashvili Equation
Shallow Water Wave Equations of Geophysical Fluid Dynamics
Computation of Solitary Wave Solutions
Direct Integration Method
The Tanh-Method
Water Wave Experiments and Observations
Future Directions
Acknowledgments
Bibliography
Primary Literature
Books and Reviews
Soliton Perturbation
Glossary
Definition of the Subject
Introduction
Methods for Soliton Solutions
Soliton Perturbation
Variational Approach
Variational Iteration Method
Homotopy Perturbation Method
Parameter-Expansion Method
Future Directions
Bibliography
Primary Literature
Books and Reviews
Solitons and Compactons
Glossary
Definition of the Subject
Introduction
Solitons
Compactons
Generalized Solitons and Compacton-Like Solutions
Coefficient of λ1
Future Directions
Cross-References
Bibliography
Primary Literature
Some Famous Papers on Solitons and Compactons
Review Article
Exp-Function Method
Parameter-Expansion Method
Nanohydrodynamics and Nano-Effect
E-Infinity Theory
Fractional-Order Differential Equations
Solitons: Historical and Physical Introduction
Glossary
Definition of the Subject
Introduction
Historical Discovery of Solitons
Physical Properties of Solitons and Associated Applications
Properties of Solitons
Solitons in Fluid Mechanics
Solitons in Nonlinear Transmission Lines
Solitons in Plasmas
Solitons in a Chain of Pendulums
Fluxons in a Josephson Tunnel Junction
Solitons in Optical Fibers
Solitons in Solid Physics
Solitons in Biology
Mathematical Methods Suitable for the Study of Solitons
Future Directions
Bibliography
Primary Literature
Books and Reviews
Solitons Interactions
Glossary
Definition of the Subject
Introduction: Key Equations, Milestones, and Methods
Integrable Equations
Milestones
Classical Soliton-Admitting Equations and Appearance of Solitons
Extended Definitions
Elastic Interactions of One-Dimensional and Line Solitons
Attraction and Repulsion
Transient Amplitude Changes, Durable Phase Shifts and Recurrence Patterns
Durable Local Amplitude Changes in Oblique Interactions of Line Solitons
Resonance
Geometry of Oblique Interactions of KP Line Solitons
Patterns
Amplitudes
Soliton Interactions in Laboratory and Nature
Ion-Acoustic Solitons
Shallow-Water and Internal Solitons
Solitons at Planetary Scale
Effects in Higher Dimensions
Optical Spatial Solitons
Coherent and Incoherent Collisions
Three-Dimensional Effects
Interactions of Vector Solitons
Applications of Line Soliton Interactions
Soliton Interactions on a Water Surface
Ship-Induced Solitons and Wave Resistance
Future Directions
Bibliography
Introduction to Solitons
Applications of Solitons, Tsunamis and Oceanographical
Glossary
Definition of the Subject
Introduction
Tsunamis
Internal Solitons
Rossby Waves and Solitons
Bore Solitons
Shallow Water Waves and KdV Type Equations
Equation of Motion: KdV Equation
Conservation of Density
Euler´s Equation
Boundary Conditions
Taylor Expansion of 훟(x,y,t) in y
Introduction of Small Parameters and δ
Perturbation Analysis
The Standard (Contemporary) Form of KdV Equation
KdV Related Integrable and Nonintegrable NLEEs
Deep Water Waves and NLS Type Equations
Tsunamis as Solitons
Basics of Tsunami Waves
The Indian Ocean Tsunami of 2004
Internal Solitons
Rossby Solitons
Bore Solitons
Future Directions
Bibliography
Primary Literature
Books and Reviews
Water Waves and the Korteweg-de Vries Equation
Glossary
Definition of the Subject
Introduction
The Euler Equation of Motion in Rectangular Cartesian and Cylindrical Polar Coordinates
Basic Equations ofWater Waves with Effects of Surface Tension
The Stokes Waves andNonlinear Dispersion Relation
Surface Gravity Waves on a Running Stream in Water
History of Russell´s Solitary Waves and Their Interactions
The Korteweg-de Vries and Boussinesq Equations
Solutions of the KdV Equation: Solitons and Cnoidal Waves
Derivation of the KdV Equation from the Euler Equations
Two-Dimensional and Axisymmetric KdV Equations
The Nonlinear SchrödingerEquation and Solitary Waves
Whitham´s Equations of Nonlinear DispersiveWaves
Whitham´s Instability Analysis of WaterWaves
The Benjamin-FeirInstability of the Stokes Water Waves
Future Directions
Bibliography
Primary Literature
Books and Reviews
Soliton Solutions for Some Nonlinear Water Wave Dynamical Models
Introduction
The Description of the AEM
Soliton Extraction
m-BBM Equation
Case I
Case II
c-DSW Equation
Case I
Case II
(3 + 1) - D e-JM Equation
Case I
Case II
Case I
Case II
Conclusion
Bibliography
Analytical Soliton Solutions for Some Nonlinear Dynamical Water Waves Models
Introduction
Applications
Application of Modified Liouville Equation
The Soliton-Like Solutions of Eq. (1)
The Trigonometric Function Solutions of Eq. (1)
The Rational Function Solutions of Eq. (1)
Application of the Symmetric Regularized Long Wave Equation
The Soliton-Like Solutions of Eq. (46)
The Trigonometric Function Solutions of Eq. (46)
The Rational Function Solution of Eq. (46)
Application of the Fourth-Order Nonlinear Ablowitz-Kaup-Newell-Segur Water Wave Equation
The Soliton-Like Solutions of Eq. (90)
The Trigonometric Function Solutions of Eq. (90)
The Rational Function Solutions of Eq. (6)
Discussion of the Results
Conclusion
Bibliography
Soliton Propagation in Solids: Advances and Applications
Introduction
Microstructure
Nonlinearity
Bulk and Surface Solitary Waves in Solids
Conclusions
Bibliography
Applications of Lump and Interaction Soliton Solutions to the Model of Liquid Crystals and Nerve Fibers
Introduction
Lump Solution
Lump-One Stripe Solution
Lump-Two Stripe Solution
Lump Periodic Solution
Periodic Cross-Kink Solutions
Figures Analysis
Concluding Remarks
Bibliography
Periodic Cross-Kink, Rogue-Waves, and Lump Interaction Soliton Solutions with Kink and Periodic Waves for Fractional Bogoyavle...
Introduction
Rogue Waves
Periodic Cross-Kink Wave Solution
Lump Two-Kink
Periodic Wave
Figures Analysis
Conclusion
Bibliography
Double Tchebyshev Spectral Tau Algorithm for Solving KdV Equation, with Soliton Application
Introduction
Orthogonal Polynomials
General Properties of Orthogonal Polynomials
Classical Jacobi Polynomials
Ultraspherical Polynomials
First Kind Chebyshev Polynomials
Second Kind Chebyshev Polynomials
Third and Fourth Kinds Chebyshev Polynomials
Spectral Methods
Galerkin Method
Tau Method
Collocation Method
Case Study
Error Estimate
Applications
Concluding Remarks
Bibliography
Index