| Preface |
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xv | |
| Acknowledgments |
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xix | |
| Author |
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xxi | |
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PART I An Example of a Unified Theory of Extreme Waves |
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Chapter 1 Lagrangian Description of Surface Water Waves |
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3 | (36) |
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1.1 The Lagrangian Form of the Hydrodynamics Equations: The Balance Equations, Boundary Conditions, and a Strongly Nonlinear Basic Equation |
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4 | (4) |
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1.1.1 Balance and State Equations |
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4 | (2) |
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1.1.2 Boundary Conditions |
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6 | (2) |
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1.1.3 A Basic Expression for the Pressure and a Basic Strongly Nonlinear Wave Equation |
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8 | (1) |
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1.2 2D Strongly Nonlinear Wave Equations for a Viscous Liquid |
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8 | (4) |
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1.2.1 The Vertical Displacement Assumption |
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9 | (1) |
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1.2.2 The 2D Airy-Type Wave Equation |
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10 | (2) |
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1.2.3 The Generation of the Green-Naghdi-Type Equation |
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12 | (1) |
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1.3 A Basic Depth-Averaged 1D Model Using a Power Approximation |
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12 | (13) |
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1.3.1 The Strongly Nonlinear Wave Equation |
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14 | (3) |
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1.3.2 Three-Speed Variants of the Strongly Nonlinear Wave Equation |
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17 | (2) |
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1.3.3 Resonant Interaction of the Gravity and Capillary Effects in a Surface Wave |
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19 | (1) |
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1.3.4 Effects of the Dispersion |
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20 | (2) |
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1.3.5 Examples of Nonlinear Wave Equations |
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22 | (3) |
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1.4 Nonlinear Equations for Gravity Waves over the Finite-Depth Ocean |
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25 | (5) |
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25 | (3) |
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1.4.2 The Gravity Waves over the Deep Ocean |
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28 | (2) |
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1.5 Models and Basic Equations for Long Waves |
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30 | (3) |
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1.6 Bottom Friction and Governing Equations for Long Extreme Waves |
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33 | (3) |
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1.7 Airy-Type Equations for Capillary Waves and Remarks to This
Chapter |
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36 | (3) |
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Chapter 2 Euler's Figures and Extreme Waves: Examples, Equations, and Unified Solutions |
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39 | (34) |
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2.1 Example of Euler's Elastica Figures |
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41 | (3) |
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2.2 Examples of Fundamental Nonlinear Wave Equations |
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44 | (1) |
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2.3 The Nonlinear Klein-Gordon Equation and Wide Spectra of Its Solutions |
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45 | (5) |
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2.3.1 The One-Dimensional Version and One-Hand Traveling Waves |
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45 | (1) |
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2.3.2 Exact Solutions of the Nonlinear Klein-Gordon Equation |
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46 | (2) |
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2.3.3 The Sine-Gordon Equation: Approximate and Exact Elastica-Like Wave Solutions |
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48 | (2) |
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2.4 Cubic Nonlinear Equations Describing Elastica-Like Waves |
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50 | (1) |
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2.5 Elastica-Like Waves: Singularities, Instabilities, Resonant Generation |
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51 | (7) |
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2.5.1 Singularities as Fields of Euler's Elastica Figures Generation |
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52 | (2) |
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2.5.2 Instabilities and Generation of Euler's Elastica Figures |
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54 | (2) |
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2.5.3 "Dangerous" Dividers and Self-Excitation of the Transresonant Waves |
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56 | (2) |
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2.6 Simple Methods for a Description of Elastica-Like Waves |
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58 | (6) |
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2.6.1 Modeling of Unidirectional Elastica-Like Waves |
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59 | (3) |
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2.6.2 The Model Equation for the Faraday Waves and Euler's Figures |
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62 | (2) |
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2.7 Nonlinear Effects on Transresonant Evolution of Euler's Figures into Particle-Waves |
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64 | (9) |
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68 | (5) |
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PART II Waves in Finite Resonators |
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Chapter 3 Generalization of d'Alembert's Solution for Nonlinear Long Waves |
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73 | (36) |
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3.1 Resonance of Traveling Surface Waves (Site Resonance) |
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73 | (5) |
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3.2 Extreme Waves in Finite Resonators |
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78 | (7) |
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3.2.1 Resonance Waves in a Gas Filling Closed Tube |
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78 | (1) |
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3.2.2 Resonant Amplification of Seismic Waves in Natural Resonators |
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78 | (4) |
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3.2.3 Topographic Effect: Extreme Dynamics of Tarzana Hill |
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82 | (3) |
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3.3 The d'Alembert-Type Nonlinear Resonant Solutions: Deformable Coordinates |
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85 | (4) |
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3.3.1 The Singular Solution of the Nonlinear Wave Equation |
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85 | (2) |
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3.3.2 The Solutions of the Wave Equation without the Singularity with Time |
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87 | (2) |
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3.3.3 Some Particular Cases of the General Solution (3.22) |
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89 | (1) |
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3.4 The d'Alembert-Type Nonlinear Resonant Solutions: Undeformable Coordinates |
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89 | (12) |
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3.4.1 The Singular Solution of the Nonlinear Wave Equations |
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90 | (4) |
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3.4.2 Resonant (Unsingular in Time) Solutions of the Wave Equation |
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94 | (3) |
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3.4.3 Special Cases of the Resonant (Unsingular with Time) Solution |
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97 | (2) |
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3.4.4 Illustration to the Theory: The Site Resonance of Waves in a Long Channel |
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99 | (2) |
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3.5 Theory of Free Oscillations of Nonlinear Wave in Resonators |
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101 | (6) |
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3.5.1 Theory of Free Strongly Nonlinear Wave in Resonators |
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102 | (3) |
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3.5.2 Comparison of Theoretical Results |
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105 | (2) |
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3.6 Conclusion on This
Chapter |
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107 | (2) |
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Chapter 4 Extreme Resonant Waves: A Quadratic Nonlinear Theory |
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109 | (38) |
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4.1 An Example of a Boundary Problem and the Equation Determining Resonant Plane Waves |
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109 | (5) |
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4.1.1 Very Small Effects of Nonlinearity, Viscosity, and Dispersion |
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110 | (1) |
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4.1.2 The Dispersion Effect on Linear Oscillations |
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111 | (1) |
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4.1.3 Fully Linear Analysis |
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112 | (2) |
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114 | (9) |
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4.2.1 Effect of Nonlinearity |
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115 | (4) |
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4.2.2 Waves Excited Very Near Boundaries of Resonant Band |
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119 | (1) |
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4.2.3 Effect of Viscosity |
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120 | (3) |
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4.3 Solutions within and Near the Shock Structure |
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123 | (3) |
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4.4 Resonant Wave Structure: Effect of Dispersion |
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126 | (5) |
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131 | (7) |
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4.5.1 Results of Calculations and Discussion |
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135 | (3) |
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4.6 Forced Vibrations of a Nonlinear Elastic Layer |
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138 | (9) |
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Chapter 5 Extreme Resonant Waves: A Cubic Nonlinear Theory |
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147 | (34) |
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5.1 Cubically Nonlinear Effect for Closed Resonators |
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150 | (17) |
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5.1.1 Results of Calculations: Pure Cubic Nonlinear Effect |
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153 | (4) |
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5.1.2 Results of Calculations: Joint Cubic and Quadratic Nonlinear Effect |
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157 | (2) |
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5.1.3 Instant Collapse of Waves Near Resonant Band End |
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159 | (1) |
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5.1.4 Linear and Cubic Nonlinear Standing Waves in Resonators |
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160 | (1) |
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5.1.5 Resonant Particles, Drops, Jets, Surface Craters, and Bubbles |
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161 | (6) |
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5.2 A Half-Open Resonator |
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167 | (7) |
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167 | (2) |
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169 | (5) |
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5.3 Scenarios of Transresonant Evolution and Comparisons with Experiments |
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174 | (1) |
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5.4 Effects of Cavitation in Liquid on Its Oscillations in Resonators |
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175 | (6) |
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Chapter 6 Spherical Resonant Waves |
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181 | (30) |
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6.1 Examples and Effects of Extreme Amplification of Spherical Waves |
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181 | (5) |
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6.2 Nonlinear Spherical Waves in Solids |
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186 | (10) |
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6.2.1 Nonlinear Acoustics of the Homogeneous Viscoelastic Solid Body |
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186 | (2) |
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6.2.2 Approximate General Solution |
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188 | (1) |
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6.2.3 Boundary Problem, Basic Relations, and Extreme Resonant Waves |
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189 | (4) |
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6.2.4 Analogy with the Plane Wave, Results of Calculations, and Discussion |
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193 | (3) |
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6.3 Extreme Waves in Spherical Resonators Filling Gas or Liquid |
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196 | (8) |
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6.3.1 Governing Equation and Its General Solution |
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196 | (1) |
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6.3.2 Boundary Conditions and Basic Equation for Gas Sphere |
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197 | (2) |
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6.3.3 Structure and Transresonant Evolution of Oscillating Waves |
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199 | (1) |
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6.3.3.1 First Scenario (C ≠ -B) |
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199 | (2) |
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6.3.3.2 Second Scenario (C = -B) |
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201 | (1) |
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202 | (2) |
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6.4 Localization of Resonant Spherical Waves in Spherical Layer |
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204 | (7) |
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Chapter 7 Extreme Faraday Waves |
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211 | (54) |
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7.1 Extreme Vertical Dynamics of Weakly Cohesive Materials |
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211 | (5) |
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7.1.1 Loosening of Surface Layers Due to Strongly Nonlinear Wave Phenomena |
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213 | (3) |
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7.2 Main Ideas of the Research |
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216 | (3) |
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7.3 Modeling Experiments as Standing Waves |
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219 | (5) |
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7.4 Modeling of Counterintuitive Waves as Travelling Waves |
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224 | (5) |
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7.4.1 Modeling of the Kolesnichenko's Experiments |
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225 | (2) |
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7.4.2 Modeling of Experiments of Bredmose et al |
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227 | (2) |
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7.5 Strongly Nonlinear Waves and Ripples |
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229 | (11) |
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7.5.1 Experiments of Lei Jiang et al. and Discussion of Them |
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229 | (4) |
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233 | (7) |
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7.6 Solitons, Oscillons, and Formation of Surface Patterns |
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240 | (5) |
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7.7 Theory and Patterns of Nonlinear Faraday Waves |
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245 | (20) |
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7.7.1 Basic Equations and Relations |
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246 | (3) |
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7.7.2 Modeling of Certain Experimental Data |
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249 | (3) |
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7.7.3 Two-Dimensional Patterns |
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252 | (4) |
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7.7.4 Historical Comments and Key Result |
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256 | (2) |
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258 | (7) |
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PART III Extreme Ocean Waves and Resonant Phenomena |
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Chapter 8 Long Waves, Green's Law and Topographical Resonance |
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265 | (36) |
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8.1 Surface Ocean Waves and Vessels |
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265 | (2) |
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8.2 Observations of the Extreme Waves |
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267 | (5) |
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272 | (3) |
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8.4 KdV-Type, Burgers-Type, Gardner-Type, and Camassa-Holm-Type Equations for the Case of the Slowly Varying Depth |
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275 | (3) |
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8.5 Model Solutions and the Green Law for Solitary Wave |
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278 | (5) |
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8.6 Examples of Coastal Evolution of the Solitary Wave |
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283 | (1) |
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8.7 Generalizations of the Green's Law |
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284 | (4) |
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8.8 Tests for Generalized Green's Law |
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288 | (7) |
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8.8.1 The Evolution of Harmonic Waves above Topographies |
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288 | (4) |
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8.8.2 The Evolution of a Solitary Wave over Trapezium Topographies |
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292 | (1) |
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8.8.3 Waves in the Channel with a Semicircular Topographies |
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293 | (2) |
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8.9 Topographic Resonances and the Euler's Elastica |
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295 | (6) |
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Chapter 9 Modeling of a Tsunami Described by Charles Darwin and Coastal Waves |
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301 | (28) |
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9.1 Darwin's Description of Tsunamis Generated by Coastal Earthquakes |
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302 | (3) |
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9.2 Coastal Evolution of Tsunami |
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305 | (8) |
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9.2.1 Effect of the Bottom Slope |
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306 | (2) |
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9.2.2 The Ocean ebb in Front of a Tsunami |
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308 | (2) |
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9.2.3 Effect of the Bottom Friction |
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310 | (3) |
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9.3 Theory of Tsunami: Basic Relations |
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313 | (5) |
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9.4 Scenarios of the Coastal Evolution of Tsunami |
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318 | (7) |
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9.4.1 Cubic Nonlinear Scenarios |
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318 | (4) |
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9.4.2 Quadratic Nonlinear Scenario |
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322 | (3) |
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9.5 Cubic Nonlinear Effects: Overturning and Breaking of Waves |
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325 | (4) |
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Chapter 10 Theory of Extreme (Rogue, Catastrophic) Ocean Waves |
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329 | (20) |
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10.1 Oceanic Heterogeneities and the Occurrence of Extreme Waves |
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330 | (3) |
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10.2 Model of Shallow Waves |
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333 | (9) |
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10.2.1 Simulation of a "Hole in the Sea" Met by the Tanker "Taganrogsky Zaliv" |
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335 | (1) |
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10.2.2 Simulation of Typical Extreme Ocean Waves as Shallow Waves |
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336 | (6) |
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10.3 Solitary Ocean Waves |
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342 | (2) |
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10.4 Nonlinear Dispersive Relation and Extreme Waves |
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344 | (3) |
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10.4.1 The Weakly Nonlinear Interaction of Many Small Amplitude Ocean Waves |
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345 | (1) |
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10.4.2 The Cubic Nonlinear Interaction of Ocean Waves and Extreme Waves Formation |
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346 | (1) |
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10.5 Resonant Nature of Extreme Harmonic Wave |
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347 | (2) |
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Chapter 11 Wind-Induced Waves and Wind-Wave Resonance |
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349 | (14) |
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11.1 Effects of Wind and Current |
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349 | (3) |
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11.2 Modeling the Effect of Wind on the Waves |
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352 | (2) |
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11.3 Relationships and Equations for Wind Waves in Shallow and Deep Water |
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354 | (2) |
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11.4 Wave Equations for Unidirectional Wind Waves |
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356 | (4) |
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11.5 The Transresonance Evolution of Coastal Wind Waves |
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360 | (3) |
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Chapter 12 Transresonant Evolution of Euler's Figures into Vortices |
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363 | (32) |
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12.1 Vortices in the Resonant Tubes |
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366 | (2) |
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12.2 Resonance Vortex Generation |
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368 | (5) |
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12.3 Simulation of the Richtmyer-Meshkov Instability Results |
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373 | (5) |
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12.4 Cubic Nonlinearity and Evolution of Waves into Vortices |
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378 | (5) |
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12.5 Remarks to Extreme Water Waves (Parts I--III) |
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383 | (12) |
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386 | (9) |
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PART IV Modeling of Particle-Waves, Slit Experiments, and the Extreme Waves in Scalar Fields |
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Chapter 13 Resonances, Euler's Figures, and Particle-Waves |
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395 | (28) |
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13.1 Scalar Fields and Euler's Figures |
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395 | (8) |
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13.1.1 Own Nonlinear Oscillations of a Scalar Field in a Resonator |
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395 | (5) |
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13.1.2 The Simplest Model of the Evolution of Euler's Figures into Periodical Particle-Wave |
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400 | (3) |
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13.2 Some Data of Exciting Experiments with Layers of Liquid |
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403 | (2) |
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13.3 Stable Oscillations of Particle-Wave Configurations |
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405 | (4) |
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13.4 Schrodinger and Klein-Gordon Equations |
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409 | (5) |
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13.5 Strongly Localized Nonlinear Sphere-Like Waves and Wave Packets |
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414 | (5) |
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13.6 Wave Trajectories, Wave Packets, and Discussion |
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419 | (4) |
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Chapter 14 Nonlinear Quantum Waves in the Light of Recent Slit Experiments |
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423 | (42) |
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423 | (4) |
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14.2 Experiments Using Different Kinds of "Slits" and the Beginning of the Discussion |
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427 | (8) |
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14.3 Explanations and Discussion of the Experimental Results |
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435 | (6) |
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441 | (4) |
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14.5 Thin Metal Layer and Plasmons as the Synchronizators |
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445 | (3) |
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14.6 Testing of thought Experiments |
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448 | (5) |
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14.7 Main thought Experiment |
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453 | (5) |
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14.8 Resonant Dynamics of Particle-Wave, Vacuum, and Universe |
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458 | (7) |
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Chapter 15 Resonant Models of Origin of Particles from Scalar Fields |
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465 | (26) |
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15.1 Basic Equation and Relations |
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466 | (2) |
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15.2 A Landscape of the Scalar Potential |
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468 | (3) |
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15.3 Effects of Interaction of Dynamic and Stationary Parts of Scalar Field: Eruption and Tunneling |
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471 | (1) |
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15.4 Description of Quantum Perturbations |
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472 | (7) |
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15.4.1 Modeling of Quantum Actions: Theory |
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473 | (1) |
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15.4.2 Modeling of Quantum Actions: Calculations |
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474 | (5) |
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15.5 Oscillations of Scalar Field and the Bose-Einstein Condensate |
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479 | (3) |
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15.6 Modelling of the Origin of the Particles |
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482 | (1) |
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15.7 Remarks and Conclusion to Part IV |
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482 | (9) |
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485 | (6) |
| Conclusion to Volume 1 |
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491 | (8) |
| Index |
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499 | |
