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E-raamat: Symmetry, Phase Modulation and Nonlinear Waves

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(University of Surrey)
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Symmetry, Phase Modulation and Nonlinear WavesSuurem pilt
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Nonlinear waves are pervasive in nature, but are often elusive when they are modelled and analysed. This book develops a natural approach to the problem based on phase modulation. It is both an elaboration of the use of phase modulation for the study of nonlinear waves and a compendium of background results in mathematics, such as Hamiltonian systems, symplectic geometry, conservation laws, Noether theory, Lagrangian field theory and analysis, all of which combine to generate the new theory of phase modulation. While the build-up of theory can be intensive, the resulting emergent partial differential equations are relatively simple. A key outcome of the theory is that the coefficients in the emergent modulation equations are universal and easy to calculate. This book gives several examples of the implications in the theory of fluid mechanics and points to a wide range of new applications.

Arvustused

'This book has been written by a well-established researcher in the field. His expertise is evidenced by the deft exposition of relatively challenging material. In that regard, one of the very useful functions of this book is its provision of a number of background mathematical techniques in Hamiltonians systems, symplectic geometry, Noether theory and Lagrangian field theory.' K. Alan Shore, Contemporary Physics 'The book is clearly written, and only the most basic knowledge of Hamiltonian and Lagrangian theories is required.' Wen-Xiu Ma, MathSciNet

Muu info

Bridges studies the origin of Kortewegde Vries equation using phase modulation and its implications in dynamical systems and nonlinear waves.
1 Introduction
1(10)
2 Hamiltonian ODEs and Relative Equilibria
11(9)
2.1 Symmetry
12(2)
2.2 Relative Equilibria
14(2)
2.3 Periodic Orbits are Relative Equilibria
16(1)
2.4 Linearization about Relative Equilibria
17(1)
2.5 Summary of Relative Equilibria
18(2)
3 Modulation of Relative Equilibria
20(6)
3.1 Modulation with Order-One Phase
22(4)
4 Revised Modulation Near a Singularity
26(17)
4.1 Jordan Chain Theory
28(2)
4.2 Fourth- and Fifth-Order Terms
30(2)
4.3 Example: Relative Equilibria with Bk = 0
32(3)
4.4 A Dynamical Systems Viewpoint
35(6)
4.5 Summary: Classification by Codimension
41(2)
5 Whitham Modulation Theory -- the Lagrangian Viewpoint
43(10)
5.1 Conservation of Wave Action
44(1)
5.2 Periodic Travelling Waves
45(2)
5.3 Averaged Lagrangian
47(1)
5.4 Whitham Theory via Fast-phase Modulation
48(3)
5.5 Remarks and Segue
51(2)
6 From Lagrangians to Multisymplectic PDEs
53(14)
6.1 Multisymplectic Hamiltonian PDEs
55(5)
6.2 Example: the Defocusing NLS Equation
60(1)
6.3 Example: a Shallow Water Boussinesq Model
61(2)
6.4 Example: the Coupled-mode Equation
63(3)
6.5 Summary
66(1)
7 Whitham Modulation Theory - the Multisymplectic Viewpoint
67(7)
7.1 Example: NLS and Linear Whitham Modulation Equations
71(1)
7.2 Breakdown of the Whitham Modulation Equations
72(2)
8 Phase Modulation and the KdV Equation
74(13)
8.1 The Modulation Ansatz
76(5)
8.2 Structure and Conservation Laws of q-KdV
81(1)
8.3 Properties and Solutions of the KdV Equation
82(1)
8.4 Example: Reduction of NLS to KdV
83(4)
9 Classical View of KdV in Shallow Water
87(9)
9.1 The Irrotational Water Wave Equations
88(2)
9.2 Reduction to a Boussinesq Equation
90(2)
9.3 Unidirectionalization via Criticality Reduction
92(4)
10 Phase Modulation of Uniform Flows and KdV
96(11)
10.1 Uniform Flows and Criticality
96(3)
10.2 Uniform Flows as Relative Equilibria
99(3)
10.3 Surface Tension
102(1)
10.4 Implicit Role of the Dispersion Relation
103(1)
10.5 Benjamin--Lighthill Theory
104(1)
10.6 KdV Equation for Deep Water Waves?
105(2)
11 Generic Whitham Modulation Theory in 2+1
107(13)
11.1 Normal Modes and Instability of Waves
112(3)
11.2 Example: 2+1 NLS and Whitham Equations
115(3)
11.3 Breakdown of the Whitham Theory in 2+1
118(2)
12 Phase Modulation in 2+1 and the KP Equation
120(18)
12.1 Averaging the Lagrangian and Wave Action
122(1)
12.2 Substituting the Modulation Ansatz
123(2)
12.3 Properties and Solutions of the KP Equation
125(1)
12.4 Dual KP Equations in 2+1
126(2)
12.5 Reflection Symmetry and Codimension
128(5)
12.6 Example: Reduction of 2+1 NLS to KP-I
133(2)
12.7 Modulation and KP in 3+1
135(2)
12.8 Restricted Modulation
137(1)
13 Shallow Water Hydrodynamics and KP
138(11)
13.1 Criticality of Uniform Flows and KP
140(2)
13.2 Reduction of a Boussinesq Model to KP
142(5)
13.3 KP for Water Waves on Infinite Depth
147(1)
13.4 Water Waves Applications of the KP Equation
147(2)
14 Modulation of Three-Dimensional Water Waves
149(20)
14.1 From Lagrangian to Multisymplectic for Water Waves
149(7)
14.2 Uniform Flows as Relative Equilibria
156(4)
14.3 Modulation of Relative Equilibria for 3D Water Waves
160(9)
15 Modulation and Planforms
169(9)
15.1 Singularities and Spatial KdV
172(2)
15.2 The Two-way Boussinesq Equation
174(2)
15.3 Planforms in the Steady RGL Equation
176(2)
16 Validity of Lagrangian-based Modulation Equations
178(7)
16.1 Validity of the SGE to NLS Reduction
180(1)
16.2 Validity of the NLS to KdV Reduction
181(3)
16.3 Reduction from an Abstract Lagrangian
184(1)
17 Non-conservative PDEs and Modulation
185(6)
17.1 Modulation and the Reaction--Diffusion Equations
186(1)
17.2 Pattern Formation and Phase Diffusion
187(1)
17.3 Gradient PDEs and Pattern Formation
188(3)
18 Phase Modulation: Extensions and Generalizations
191(3)
Appendix A Supporting Calculations: Fourth- and Fifth-Order Terms 194(4)
Appendix B Derivatives of a Family of Relative Equilibria 198(3)
Appendix C Bk and the Spectral Problem 201(4)
Appendix D Reducing Dispersive Conservation Laws to KdV 205(3)
Appendix E Advanced Topics in Multisymplecticity 208(8)
References 216(10)
Index 226
Thomas J. Bridges is currently Professor of Mathematics at the University of Surrey. He has been researching the theory of nonlinear waves for over 25 years. He is co-editor of the volume Lectures on the Theory of Water Waves (Cambridge, 2016) and he has over 140 published papers on such diverse topics as multisymplectic structures, Hamiltonian dynamics, ocean wave energy harvesting, geometric numerical integration, stability of nonlinear waves, the geometry of the Hopf bundle, theory of water waves and phase modulation.
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