In the 1960s and 1970s, Benjamin and Ono proposed a type of development equation of Hilbert transformation with singularity in [107] and [113] respectively.
Among them, α and β are constants, and H is the Hilbert transformation, which we call The Benjamin-Ono equation, referred to as the BO equation. The BO equation is used to describe the motion of water waves in deep water, and also describes the three-layer optical resonance in optical media. The BO equation is related to the Korteweg-deVries (KdV) equation that describes shallow water waves and limited depth water waves. The equations together become an important dispersive water wave equation, and there is a close connection between the three. When the parameter describing the fluid depth δ→∞, the finite depth water wave equation is close to the BO equation; when the parameter describing the fluid depth δ→0 When , the finite depth water wave equation approximates the KdV equation. It can be proved that the BO equation is an integrable system, with solitons . Different from the bell-shaped solitons of the KdV equation, the BO equation has finite fractional algebraic solitons.
In 1986, Zhou Yulin and Guo Boling proved for the first time the existence and uniqueness of the overall smooth solution of the BO equation. Since then, many famous scholars at home and abroad have done a lot of research on the BO equation, such as T. Tao, C. E. Kenig, etc., and obtained the L2 space A series of results such as the large initial value global solution of Space, make full use of tools such as harmonic analysis .
The purpose of this book is to comprehensively introduce some important mathematical theories, research methods and research such as the BO equation and the finite depth water wave equation in a concise, clear and easy-to-understand form. Results, as well as some of the author's research results, including the existence, uniqueness, low regularity, asymptotic properties of the global solution in energy space and Bourgain space, as well as the orbital stability and asymptotic stability of the solitary wave solution. We It is hoped that the publication of this book will help mathematics and physics researchers, especially young researchers, to have a general understanding of the BO equation. If you are interested in these, you can consult the relevant literature listed in this book. Carry out new research work on the BO equation faster and more deeply.
[107] Benjamin T B. Internal waves of permanent form in fluids of great depth. J. Fluid Mech., 1967, 29: 559-592.
[113 ] Ono H. Algebraic solitary waves in stratieduids fluids. J. Phys. Soc. Japan, 1975, 39: 1082-1091.
The Benjamin-Ono equation and its strange wave solution in deep water
Guo Boling et al.
Beijing: Science Press
ISBN 978-7-03-071508-1
Editors: Li Xin, Li Ping
The Benjamin-Ono (BO) equation in deep water is a very important type of nonlinear dispersion equation with a wide range of physical background and application background. This type of equation has a class of algebraic solitons with finite fractions and is an integrable system. This book gives the physical background of this type of equation and explains its strange wave solution. It focuses on the mathematical theory of several important types of BO equations, including the existence, uniqueness and low cost of the overall solution in energy space and Bourgain space. Regularity etc. At the same time, this book studies the generalized solutions of the medium-depth water wave equation, the asymptotic and limiting properties of the solutions, the explosive properties of the solutions of the generalized KP equation and the two-dimensional BO equation, and introduces the isolation of the BO equation using the methods of stability theory and spectral analysis. Orbital and asymptotic stability of wave solutions.
In the 1960s and 1970s, Benjamin and Ono proposed a type of development equation of Hilbert transformation with singularity in [107] and [113] respectively.
Among them, α and β are constants, and H is the Hilbert transformation, which we call The Benjamin-Ono equation, referred to as the BO equation. The BO equation is used to describe the motion of water waves in deep water, and also describes the three-layer optical resonance in optical media. The BO equation is related to the Korteweg-deVries (KdV) equation that describes shallow water waves and limited depth water waves. The equations together become an important dispersive water wave equation, and there is a close connection between the three. When the parameter describing the fluid depth δ→∞, the finite depth water wave equation is close to the BO equation; when the parameter describing the fluid depth δ→0 When , the finite depth water wave equation approximates the KdV equation. It can be proved that the BO equation is an integrable system, with solitons . Different from the bell-shaped solitons of the KdV equation, the BO equation has finite fractional algebraic solitons.
In 1986, Zhou Yulin and Guo Boling proved for the first time the existence and uniqueness of the overall smooth solution of the BO equation. Since then, many famous scholars at home and abroad have done a lot of research on the BO equation, such as T. Tao, C. E. Kenig, etc., and obtained the L2 space A series of results such as the large initial value global solution of Space, make full use of tools such as harmonic analysis .
The purpose of this book is to comprehensively introduce some important mathematical theories, research methods and research such as the BO equation and the finite depth water wave equation in a concise, clear and easy-to-understand form. Results, as well as some of the author's research results, including the existence, uniqueness, low regularity, asymptotic properties of the global solution in energy space and Bourgain space, as well as the orbital stability and asymptotic stability of the solitary wave solution. We It is hoped that the publication of this book will help mathematics and physics researchers, especially young researchers, to have a general understanding of the BO equation. If you are interested in these, you can consult the relevant literature listed in this book. Carry out new research work on the BO equation faster and more deeply.
[107] Benjamin T B. Internal waves of permanent form in fluids of great depth. J. Fluid Mech., 1967, 29: 559-592.
[113 ] Ono H. Algebraic solitary waves in stratieduids fluids. J. Phys. Soc. Japan, 1975, 39: 1082-1091.
The Benjamin-Ono equation and its strange wave solution in deep water
Guo Boling et al.
Beijing: Science Press
ISBN 978-7-03-071508-1
Editors: Li Xin, Li Ping
The Benjamin-Ono (BO) equation in deep water is a very important type of nonlinear dispersion equation with a wide range of physical background and application background. This type of equation has a class of algebraic solitons with finite fractions and is an integrable system. This book gives the physical background of this type of equation and explains its strange wave solution. It focuses on the mathematical theory of several important types of BO equations, including the existence, uniqueness and low cost of the overall solution in energy space and Bourgain space. Regularity etc. At the same time, this book studies the generalized solutions of the medium-depth water wave equation, the asymptotic and limiting properties of the solutions, the explosive properties of the solutions of the generalized KP equation and the two-dimensional BO equation, and introduces the isolation of the BO equation using the methods of stability theory and spectral analysis. Orbital and asymptotic stability of wave solutions.
Table of contents quick overview
Preface
Chapter 1 The physical background of the Benjamin-Ono equation and its strange wave solution 1
1.1 Introduction 1
1.2 The derivation of the Benjamin-Ono equation and its solitary wave solution 1
1.3 The underlying equation (0≤y<>
1.4 Matching of the upper equation (y≥h0) and y=h0 6
1.5 Regarding the conservation law of equation (1.4.51) 8
1.6 The steady traveling wave of equation (1.4.51) 9
1.7 Finite depth fluid Solitary wave 11
Chapter 2 Smooth solution to the initial value problem of the Benjamin-Ono equation 13
2.1 Generalized Benjamin-Ono equation with diffusion term 13
2.2 Priori estimation 16
2.3 Generalized solution 21
Chapter 3 The overall Benjamin-Ono equation Low canonical solution 23
3.1 Introduction 23
3.2 Current research status of well-posedness of Benjamin-Ono equation 23
3.3 Large initial value global solution of Benjamin-Ono equation in L2 space 25
3.4 Gauge transformation 27
3.5 Construction of work space 30
3. 6 Properties of space Zk 33
3.7 Linear estimation 39
3.8 Local L2 estimation 44
3.9 Bilinear estimation Low×High→High 50
3.10 Bilinear estimation High×High→Low 61
3.11 Multiplication of smooth bounded functions Sub-estimation 67
3.12 Proof of Theorem 3.3.1 76
Chapter 4 Hs solution of KdV-BO-Hirota equation 90
4.1 Introduction 90
4.2 Preliminary knowledge 92
4.3 Local results 94
4.4 Hirota equation in Hs(1≤s≤ 2) The overall solution on 96
Chapter 5 Hs solution of BO long and short wave equations 97
5.1 Introduction 97
5.2 Some estimation lemma 98
5.3 Nonlinear estimation 101
Chapter 6 Generalized solution of medium depth water wave equation 111
6.1 Introduction111
6.2 Some properties of the singular integral operator G(u)112
6.3 Solvability of equation (6.1.6) for α0 116
6.4 Existence of local solution of equation (6.3.13), α=0 118
6.5 Overall solvability of equation (6.3.13) 120
Chapter 7 Asymptotic properties of solutions to the medium depth water wave equation 126
7.1 Introduction 126
7.2 Some lemmas 127
7.3 Linear estimation 131
7.4 Attenuation of nonlinear problems Estimation 137
Chapter 8 Limiting properties of the medium depth water wave equation 141
8.1 Introduction 141
8.2 Global well-posedness of the generalized finite depth water wave equation 141
8.3 Linear estimation 146
8.4 Small initial value global well-posedness 161
8.4.1 Construction of work space E 161
8.4.2 Proof of Theorem 8.2.6164
8.4.3 Proof of Theorem 8.2.5171
8.5 Limit behavior of the solution176
8.5.1 Regularity of the solution176
8.5.2 Approximation of the solution to the KdV equation when δ→0178
8.5.3 When Approximation of solutions to Benjamin-Ono equation when δ→∞183
Chapter 9 Explosion of solutions of generalized KP equation and two-dimensional Benjamin-Ono equation 188
9.1 Introduction 188
9.2 Local conclusion 189
9.3 Explosion conclusion 194
Chapter 10 Generalized stochastic Benjamin -Initial value problem of Ono equation 200
10.1 Introduction 200
10.2 Preliminary knowledge 203
10.3 Bilinear estimation 206
10.4 Trilinear estimation 210
10.5 Local well-posedness 213
10.6 Proof of Theorem 10.1.2 214
Chapter 11 KdV- Low regularity problem of BO equation 225
11.1 Introduction 225
11.2 Preliminary knowledge 227
11.3 Local solution when l=2 231
11.4 Proof of Theorem 11.1.4 237
Chapter 12 Orbital stability of solitary wave solution of Benjamin-Ono equation 239
12.1 Existence of solitary wave solution 239
12.2 Main results 241
Chapter 13 Asymptotic stability of solitary wave solution of Benjamin-Ono equation 245
13.1 Introduction 245
13.2 Some monotonicity results 247
13.2.1 Preparatory work 247
13.2.2 Modulation Lemma 248
13.2.3 Monotonicity of u(t) 249
13.2.4 Monotonicity of η(t) 258
13.3 Linear Liouville theorem 262
13.3.1 Prove theorem 13.3.1 assuming quadratic form positive definite 263
13.3.2 dual Positive definite quadratic form of the problem276
13.4 Asymptotic stability280
13.4.1 Proof of Theorem 13.1.1280
13.4.2 Proof of Theorem 13.1.2284
13.4.3 Proof of Note 13.1.3292
13.5 Case of multiple solitons 294
13.5.1 Generalization of stability theory 295
13.5.2 Generalization of the proof of Theorem 13.5.1 296
13.6 Weak convergence and well-posed results 299
13.6.1 Weak convergence 299
13.6.2 Well-posed results of nonlinear BO equations 300
Reference 310
(edited in this article : Wang Fang)
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