Solitons

Description

This newly updated volume of the  Encyclopedia of Complexity and Systems Science (ECSS) presents several mathematical models that describe this physical phenomenon, including the famous non-linear equation Korteweg-de-Vries (KdV) that represents the canonical form of solitons. Also, there exists a class of nonlinear partial differential equations that led to solitons, e.g., Kadomtsev-Petviashvili (KP), Klein-Gordon (KG), Sine-Gordon (SG), Non-Linear Schrödinger (NLS), Korteweg-de-Vries Burger’s (KdVB), etc. Different linear mathematical methods can be used to solve these models analytically, such as the Inverse Scattering Transformation (IST), Adomian Decomposition Method, Variational Iteration Method (VIM), Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM). Other non-analytic methods use the computational techniques available in such popular mathematical packages as Mathematica, Maple, and MATLAB. The main purpose of this volume is to provide physicists, engineers, and their students with the proper methods and tools to solve the soliton equations, and to discover the new possibilities of using solitons in multi-disciplinary areas ranging from telecommunications to biology, cosmology, and oceanographic studies. 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