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