Nonlinear waves and solitons on contours and closed surfaces / / Andrei Ludu

• Autore: Ludu Andrei
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Edizione: Third edition.
• Descrizione fisica: 1 online resource (583 pages)
• Disciplina: 514.32
• Soggetto topico: Compact spaces
• Note generali: Includes index.
• Nota di contenuto: Intro -- Foreword -- Preface to the Third Edition -- Preface to the Second Edition -- Preface to the First Edition -- Contents -- Symbols -- 1 Introduction -- 1.1 Intuitive Introduction to Nonlinear Waves and Solitons -- 1.2 Integrability -- 1.3 Algebraic and Geometric Approaches -- 1.4 A List of Useful Derivatives in Finite Dimensional Spaces -- References -- Part I Mathematical Prerequisites -- 2 Topology and Algebra -- 2.1 What Is Topology -- 2.1.1 Topological Spaces and Separation -- 2.1.2 Compactness and Weierstrass-Stone Theorem -- 2.1.3 Connectedness and Homotopy -- 2.1.4 Separability and Metric Spaces -- 2.2 Elements of Homology -- 2.3 Group Action -- References -- 3 Vector Fields, Differential Forms, and Derivatives -- 3.1 Manifolds and Maps -- 3.2 Differential and Vector Fields -- 3.3 Existence and Uniqueness Theorems: Differential Equation Approach -- 3.4 Existence and Uniqueness Theorems: Flow Box Approach -- 3.5 Compact Supported Vector Fields -- 3.6 Differential Forms and the Lie Derivative -- 3.7 Differential Systems, Integrability and Invariants -- 3.8 Poincaré Lemma -- 3.9 Fiber Bundles and Covariant Derivative -- 3.9.1 Principal Bundle and Frames -- 3.9.2 Connection Form and Covariant Derivative -- 3.10 Tensor Analysis -- 3.11 The Mixed Covariant Derivative -- 3.12 Curvilinear Orthogonal Coordinates -- 3.13 Special Two-Dimensional Nonlinear Orthogonal Coordinates -- 3.14 Problems -- References -- 4 The Importance of the Boundary -- 4.1 The Power of Compact Boundaries: Representation Formulas -- 4.1.1 Representation Formula for n=1: Taylor Series -- 4.1.2 Representation Formula for n=2: Cauchy Formula -- 4.1.3 Representation Formula for n=3: Green Formula -- 4.1.4 Representation Formula in General: Stokes Theorem -- 4.2 Comments and Examples -- References -- Part II Curves and Surfaces -- 5 Geometry of Curves. ; 5.1 Elements of Differential Geometry of Curves -- 5.2 Closed Curves -- 5.3 Curves Lying on a Surface -- 5.4 Problems -- References -- 6 Geometry of Surfaces -- 6.1 Elements of Differential Geometry of Surfaces -- 6.2 Covariant Derivative and Connections -- 6.3 Geometry of Parameterized Surfaces Embedded in mathbbR3 -- 6.3.1 Christoffel Symbols and Covariant Differentiation for Hybrid Tensors -- 6.4 Compact Surfaces -- 6.5 Surface Differential Operators -- 6.5.1 Surface Gradient -- 6.5.2 Surface Divergence -- 6.5.3 Surface Laplacian -- 6.5.4 Surface Curl -- 6.5.5 Integral Relations for Surface Differential Operators -- 6.5.6 Applications -- 6.6 Problems -- References -- 7 Motion of Curves and Solitons -- 7.1 Kinematics of Two-Dimensional Curves -- 7.2 Mapping Two-Dimensional Curve Motion into Nonlinear Integrable Systems -- 7.3 The Time Evolution of Length and Area -- 7.4 Cartan Theory of Three-Dimensional Curve Motion -- 7.5 Kinematics of Three-Dimensional Curves -- 7.6 Mapping Three-Dimensional Curve Motion into Nonlinear Integrable Systems -- 7.7 Problems -- References -- 8 Theory of Motion of Surfaces -- 8.1 Differential Geometry of Surface Motion -- 8.2 Coordinates and Velocities on a Fluid Surface -- 8.3 Kinematics of Moving Surfaces -- 8.4 Dynamics of Moving Surfaces -- 8.5 Boundary Conditions for Moving Fluid Interfaces -- 8.6 Dynamics of the Fluid Interfaces -- 8.7 Problems -- References -- Part III Solitons and Nonlinear Waves on Closed Curves and Surfaces -- 9 Kinematics of Fluids -- 9.1 Lagrangian Verses Eulerian Frames -- 9.1.1 Introduction -- 9.1.2 Geometrical Picture for Lagrangian Verses Eulerian -- 9.2 Fluid Fiber Bundle -- 9.2.1 Introduction -- 9.2.2 Motivation for a Geometrical Approach -- 9.2.3 The Fiber Bundle -- 9.2.4 Fixed Fluid Container -- 9.2.5 Free Surface Fiber Bundle. ; 9.2.6 How Does the Time Derivative of Tensors Transform from Euler to Lagrange Frame? -- 9.3 Path Lines, Stream Lines, and Particle Contours -- 9.4 Eulerian-Lagrangian Description for Moving Curves -- 9.5 The Free Surface -- 9.6 Equation of Continuity -- 9.6.1 Introduction -- 9.6.2 Solutions of the Continuity Equation on Compact Intervals -- 9.7 Problems -- References -- 10 Hydrodynamics -- 10.1 Momentum Conservation: Euler and Navier-Stokes Equations -- 10.2 Boundary Conditions -- 10.3 Circulation Theorem -- 10.4 Surface Tension -- 10.4.1 Physical Problem -- 10.4.2 Minimal Surfaces -- 10.4.3 Application -- 10.4.4 Isothermal Parametrization -- 10.4.5 Topological Properties of Minimal Surfaces -- 10.4.6 General Condition for Minimal Surfaces -- 10.4.7 Surface Tension for Almost Isothermal Parametrization -- 10.5 Special Fluids -- 10.6 Representation Theorems in Fluid Dynamics -- 10.6.1 Helmholtz Decomposition Theorem in mathbbR3 -- 10.6.2 Decomposition Formula for Transversal Isotropic Vector Fields -- 10.6.3 Solenoidal-Toroidal Decomposition Formulas -- 10.7 Problems -- References -- 11 Nonlinear Surface Waves in One Dimension -- 11.1 KdV Equation Deduction for Shallow Waters -- 11.2 Smooth Transitions Between Periodic and Aperiodic Solutions -- 11.3 Modified KdV Equation and Generalizations -- 11.4 Hydrodynamic Equations Involving Higher-Order Nonlinearities -- 11.4.1 A Compact Version for KdV -- 11.4.2 Small Amplitude Approximation -- 11.4.3 Dispersion Relations -- 11.4.4 The Full Equation -- 11.4.5 Reduction of GKdV to Other Equations and Solutions -- 11.4.6 The Finite Difference Form -- 11.5 Boussinesq Equations on a Circle -- References -- 12 Nonlinear Surface Waves in Two Dimensions -- 12.1 Geometry of Two-Dimensional Flow -- 12.2 Two-Dimensional Nonlinear Equations -- 12.3 Two-Dimensional Fluid Systems with Moving Boundary. ; 12.4 Oscillations in Two-Dimensional Liquid Drops -- 12.5 Contours Described by Quartic Closed Curves -- 12.6 Nonlinear Waves in Rotating Leidenfrost Drops -- References -- 13 Dynamics of Two-Dimensional Fluid in Bounded Domain via Conformal Variables (A. Chernyavsky and S. Dyachenko) -- 13.1 Introduction -- 13.2 Mechanics of Droplet and the Conformal Map -- 13.2.1 The Hamiltonian, Momentum and Angular Momentum -- 13.2.2 The Center of Mass -- 13.3 The Complex Equations of Motion -- 13.3.1 Kinematic Equation -- 13.3.2 Dynamic Condition -- 13.4 Traveling Waves Around a Disk -- 13.5 Linear Waves -- 13.6 Numerical Simulation -- 13.7 Series Solution -- 13.8 Nonlinear Waves -- 13.9 Conclusion -- References -- 14 Nonlinear Surface Waves in Three Dimensions -- 14.1 Oscillations of Inviscid Drops: The Linear Model -- 14.1.1 Drop Immersed in Another Fluid -- 14.1.2 Drop with Rigid Core -- 14.1.3 Moving Core -- 14.1.4 Drop Volume -- 14.2 Oscillations of Viscous Drops: The Linear Model -- 14.2.1 Model 1 -- 14.3 Nonlinear Three-Dimensional Oscillations of Axisymmetric Drops -- 14.3.1 Nonlinear Resonances in Drop Oscillation -- 14.4 Other Nonlinear Effects in Drop Oscillations -- 14.5 Solitons on the Surface of Liquid Drops -- 14.6 Problems -- References -- 15 Other Special Nonlinear Compact Systems -- 15.1 Solitons on Interfaces of Layered Fluid Droplet (Written by A. S. Carstea) -- 15.2 Nonlinear Compact Shapes and Collective Motion -- 15.3 The Hamiltonian Structure for Free Boundary Problems on Compact Surfaces -- References -- Part IV Physical Nonlinear Systems at Different Scales -- 16 Filaments, Chains, and Solitons -- 16.1 Vortex Filaments -- 16.1.1 Gas Dynamics Filament Model and Solitons -- 16.1.2 Special Solutions -- 16.1.3 Integration of Serret-Frenet Equations for Filaments -- 16.1.4 The Riccati Form of the Serret-Frenet Equations. ; 16.2 Soliton Solutions on the Vortex Filament -- 16.2.1 Constant Torsion Vortex Filaments -- 16.2.2 Vortex Filaments and the Nonlinear Schrödinger Equation -- 16.3 Closed Curves Solitons -- 16.4 Nonlinear Dynamics of Stiff Chains -- 16.5 Problems -- References -- 17 Solitons on the Boundaries of Microscopic Systems -- 17.1 Solitons as Elementary Particles -- 17.2 Quantization of Solitons on a Closed Contour and Instantons -- 17.3 Clusters as Solitary Waves on the Nuclear Surface -- 17.4 Nonlinear Schrödinger Equation Solitons on Quantum … -- 17.5 Solitons and Quasimolecular Structure -- 17.6 Soliton Model for Heavy Emitted Nuclear Clusters -- 17.7 Quintic Nonlinear Schrödinger Equation for Nuclear Cluster Decay -- 17.8 Contour Solitons in the Quantum Hall Liquid -- References -- 18 Nonlinear Contour Dynamics in Macroscopic Systems -- 18.1 Plasma Vortex -- 18.1.1 Effective Surface Tension in Magnetohydrodynamics and Plasma Systems -- 18.1.2 Trajectories in Magnetic Field Configurations -- 18.1.3 Magnetic Surfaces in Static Equilibrium -- 18.2 Elastic Spheres -- 18.3 Curvature Dependent Nonlinear Diffusion on Closed Surfaces -- 18.4 Nonlinear Evolution of Oscillation Modes in Neutron Stars -- References -- 19 Mathematical Appendix -- 19.1 Differentiable Manifolds -- 19.2 Riccati Equation -- 19.3 Special Functions -- 19.4 One-Soliton Solutions for the KdV, MKdV, and Their Combination -- 19.5 Scaling and Nonlinear Dispersion Relations1 -- References -- Index.
• Titolo autorizzato: Nonlinear Waves and Solitons on Contours and Closed Surfaces
• ISBN: 9783031146411; 9783031146404
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996499864303316
• Lo trovi qui: Univ. di Salerno
• Serie: Springer Series in Synergetics