Solitons : a volume in the encyclopedia of complexity and systems science / / Mohamed Atef Helal, editor

• Record Nr.: UNISA996499863103316
• Pubbl/distr/stampa: New York, New York : , : Springer, , 2022
• ISBN: 9781071624579; 9781071624562
• Edizione: [Second edition.]
• Descrizione fisica: 1 online resource (483 pages)
• Collana: Encyclopedia of complexity and systems science series
• Disciplina: 530.124
• Soggetti: Solitons
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- 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.