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Camasa-Holm方程(英文版) 版权信息
- ISBN：9787030595577
- 条形码：9787030595577 ; 978-7-03-059557-7
- 装帧：一般胶版纸
- 册数：暂无
- 重量：暂无
- 所属分类：>
Camasa-Holm方程(英文版) 内容简介
Camassa-Holm方程是一类十分重要而又特别的新型浅水波方程，有广泛的应用背景。该类方程存在一类尖峰孤立子，并且它是接近可积的,具有双哈密顿结构和Lax对。本书给出该类方程的物理背景并阐述它的接近可积性。对该类方程的行波解作分类，获得多种奇异孤立波解；给出该类方程的谱图理论和散射数据；利用反散射方法，给出该类方程的多孤立子解。获得该类方程的整体强解的存在性及整体弱解的存在性；得到该类方程柯西问题的局部适定性；研究它们的blow-up问题以及尖峰孤立子解的轨道稳定性。本书同时研究含尖峰孤立子的Degasperis-Procesi方程及b族方程，研究前一类方程激波的形成及动力学分析，给出b族方程的水波结构和非线性平衡关系，对Degasperis-Procesi方程的适定性给出具体证明。
Camasa-Holm方程(英文版) 目录
Contents
Chapter 1 Physical Backgrounds and Complete Integrability of the Camassa-Holm Equation 1
1.1 Physical backgrounds of the Camassa-Holm equation 1
1.2 The complete integrability of the Camassa-Holm equation 9
1.3 Experimental observation and applications of solitons 17
References 18
Chapter 2 Traveling Wave Solutions of the Camassa-Holm Equation 33
2.1 Introduction 33
2.2 Notations 34
2.3 Weak form 36
2.4 Several types of traveling wave solutions 37
2.5 The proof Theorem 2.4.1 43
2.6 The correlation of parameters 59
2.7 Wave length 63
2.8 Explicit formulae of peakon 66
References 68
Chapter 3 The Scattering and Inverse Scattering of the Camassa-Holm Equation 71
3.1 Scattering of the Camassa-Holm equation 71
3.1.1 Introduction 71
3.1.2 Spectral graph theory 72
3.1.3 The scattering problem 82
3.2 The solutions of Camassa-Holm equation 89
3.2.1 Introduction 89
3.2.2 Summary of the process 89
3.2.3 Summary of solving process 93
3.2.4 Solitary wave solutions 93
3.2.5 2-soliton solutions 97
3.2.6 Examples and properties of 2-soliton solutions 102
3.2.7 3-soliton solutions 106
3.2.8 Summary 109
References 111
Chapter 4 Well-posedness of the Camassa-Holm Equation 113
4.1 Global existence of strong solutions 113
4.1.1 The existence of local solutions 113
4.1.2 The existence of global solutions 120
4.2 The existence of global weak solutions 128
4.3 The local well posedness of the Cauchy problem to the Camassa-Holm equation in Hs (s > 2/3) 138
4.4 Blow-up phenomena of the Camassa-Holm equation 145
4.5 The orbital stability of peakon solutions 153
References 157
Chapter 5 Formation and Dynamics Analysis of Shock Wave of the Degasperis-Procesi Equation 159
5.1 Introduction 159
5.1.1 Peakons 159
5.1.2 Generalized weak solutions 162
5.2 The shock wave of the DP equation 164
5.3 Peakon， anti-peakon and the formation of shock waves 172
5.4 Shock wave dynamics 184
5.5 Concluding remarks 190
References 191
Chapter 6 Water Wave Structure and Nonlinear Equilibrium of 6-Family Nonlinear Shallow Water Wave Equation 194
6.1 Introduction 195
6.1.1 6-family shallow water wave equation 195
6.1.2 Outline of the paper 196
6.2 History and general properties of the 6-equation 197
6.2.1 Discrete symmetries: reversibility， parity， and signature 199
6.2.2 Lagrangian representation 199
6.2.3 Preservation of the norm ||m||L1/b， 0≤b≤1 200
6.2.4 Lagrangian representation for integer b 201
6.2.5 Reversibility and Galilean covariance 202
6.2.6 Integral momentum conservation 202
6.3 Traveling waves and generalized functions 203
6.3.1 The case of b = 0 203
6.3.2 The case of b ≠ 0 205
6.3.3 The case of b > 0 206
6.3.4 The case of b 0 214
6.4.1 Pulson interactions for b = 2 215
6.4.2 Peakon interactions for b = 2 and b = 3: numerical results 215
6.4.3 Pulson-pulson interactions for b > 0 and symmetric g 217
6.4.4 Pulson-antipulson interactions for b> 1 and symmetric g 220
6.4.5 Specializing pulsons to peakons for b = 2 and b = 3 222
6.5 Peakons of width a for arbitrary b 223
6.5.1 The slope dynamics of peakons: inflection points and the steepening lemma when 1 6.5.2 The case of 0≤b≤1 226
6.6 Adding viscosity to peakon dynamics 226
6.6.1 Burgers-a^ equation: analytical estimates 228
6.6.2 The traveling waves of Burgers-αβ equation for P(3 - b) = 1 and v = 0 230
6.7 The peakons under (6.1.1) adding viscosity and evolution of (6.1.2) Burgers-αβ 231
6.7.1 The fate of peakons under adding viscosity 231
6.7.2 The fate of peakons under Burgers-ap evolution 238
6.8 Numerical results for peakon scattering and initial value problems 241
6.8.1 Peakon initial value problems 241
6.8.2 Description of our numerical methods 243
6.9 Conclusions 245
References 247
Chapter 7 The Degasperis-Procesi Equation 249
7.1 Introduction 249
7.2 Local well-posedness 253
7.3 Blow up 255
7.4 Global strong solutions 259
7.5 Global weak solutions 263
7.6 Recent results and problems 278
References 284
Chapter 1 Physical Backgrounds and Complete Integrability of the Camassa-Holm Equation 1
1.1 Physical backgrounds of the Camassa-Holm equation 1
1.2 The complete integrability of the Camassa-Holm equation 9
1.3 Experimental observation and applications of solitons 17
References 18
Chapter 2 Traveling Wave Solutions of the Camassa-Holm Equation 33
2.1 Introduction 33
2.2 Notations 34
2.3 Weak form 36
2.4 Several types of traveling wave solutions 37
2.5 The proof Theorem 2.4.1 43
2.6 The correlation of parameters 59
2.7 Wave length 63
2.8 Explicit formulae of peakon 66
References 68
Chapter 3 The Scattering and Inverse Scattering of the Camassa-Holm Equation 71
3.1 Scattering of the Camassa-Holm equation 71
3.1.1 Introduction 71
3.1.2 Spectral graph theory 72
3.1.3 The scattering problem 82
3.2 The solutions of Camassa-Holm equation 89
3.2.1 Introduction 89
3.2.2 Summary of the process 89
3.2.3 Summary of solving process 93
3.2.4 Solitary wave solutions 93
3.2.5 2-soliton solutions 97
3.2.6 Examples and properties of 2-soliton solutions 102
3.2.7 3-soliton solutions 106
3.2.8 Summary 109
References 111
Chapter 4 Well-posedness of the Camassa-Holm Equation 113
4.1 Global existence of strong solutions 113
4.1.1 The existence of local solutions 113
4.1.2 The existence of global solutions 120
4.2 The existence of global weak solutions 128
4.3 The local well posedness of the Cauchy problem to the Camassa-Holm equation in Hs (s > 2/3) 138
4.4 Blow-up phenomena of the Camassa-Holm equation 145
4.5 The orbital stability of peakon solutions 153
References 157
Chapter 5 Formation and Dynamics Analysis of Shock Wave of the Degasperis-Procesi Equation 159
5.1 Introduction 159
5.1.1 Peakons 159
5.1.2 Generalized weak solutions 162
5.2 The shock wave of the DP equation 164
5.3 Peakon， anti-peakon and the formation of shock waves 172
5.4 Shock wave dynamics 184
5.5 Concluding remarks 190
References 191
Chapter 6 Water Wave Structure and Nonlinear Equilibrium of 6-Family Nonlinear Shallow Water Wave Equation 194
6.1 Introduction 195
6.1.1 6-family shallow water wave equation 195
6.1.2 Outline of the paper 196
6.2 History and general properties of the 6-equation 197
6.2.1 Discrete symmetries: reversibility， parity， and signature 199
6.2.2 Lagrangian representation 199
6.2.3 Preservation of the norm ||m||L1/b， 0≤b≤1 200
6.2.4 Lagrangian representation for integer b 201
6.2.5 Reversibility and Galilean covariance 202
6.2.6 Integral momentum conservation 202
6.3 Traveling waves and generalized functions 203
6.3.1 The case of b = 0 203
6.3.2 The case of b ≠ 0 205
6.3.3 The case of b > 0 206
6.3.4 The case of b 0 214
6.4.1 Pulson interactions for b = 2 215
6.4.2 Peakon interactions for b = 2 and b = 3: numerical results 215
6.4.3 Pulson-pulson interactions for b > 0 and symmetric g 217
6.4.4 Pulson-antipulson interactions for b> 1 and symmetric g 220
6.4.5 Specializing pulsons to peakons for b = 2 and b = 3 222
6.5 Peakons of width a for arbitrary b 223
6.5.1 The slope dynamics of peakons: inflection points and the steepening lemma when 1 6.5.2 The case of 0≤b≤1 226
6.6 Adding viscosity to peakon dynamics 226
6.6.1 Burgers-a^ equation: analytical estimates 228
6.6.2 The traveling waves of Burgers-αβ equation for P(3 - b) = 1 and v = 0 230
6.7 The peakons under (6.1.1) adding viscosity and evolution of (6.1.2) Burgers-αβ 231
6.7.1 The fate of peakons under adding viscosity 231
6.7.2 The fate of peakons under Burgers-ap evolution 238
6.8 Numerical results for peakon scattering and initial value problems 241
6.8.1 Peakon initial value problems 241
6.8.2 Description of our numerical methods 243
6.9 Conclusions 245
References 247
Chapter 7 The Degasperis-Procesi Equation 249
7.1 Introduction 249
7.2 Local well-posedness 253
7.3 Blow up 255
7.4 Global strong solutions 259
7.5 Global weak solutions 263
7.6 Recent results and problems 278
References 284
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Camasa-Holm方程(英文版) 节选
Chapter 1 Physical Backgrounds and Complete Integrability of the Camassa-Holm Equation 1.1 Physical backgrounds of the Camassa-Holm equation In the shallow water theory， a fairly wide range of wave equation and equa-tions describing weakly nonlinear effects can be regarded as the Korteweg-de Vries (KdV) equation under the assumptions that wavelengths approximate and amplitudes are small and finite. For example: (1) The motion of magnetic fluid wave of cold plasma; (2) Vibration of anharmonic lattice; (3) Ion-asoustic wave of plasma; (4) Longitudinal dispersion wave in elastic rod; (5) The pressure wave motion of the mixture of liquid and gas; (6) The rotation of fluid at the bottom of a tube; (7) The thermal excitation radiation of a phonon wave packet of the non-linear lattice at low temperature. Different degrees of approximation can yield different nonlinear partial diffe-rential equations that are completely integrable in the shallow wave theory. These equations have soliton solutions， i.e.， they tend to be flexibly given constants at infinity in space. They don’t disappear after colliding with each other and have no or only slight changes in wave form and velocity， and they show behaviors similar to particle scattering. In this section， we will use the Hamiltonian method to derive a new type of shallow water equation， i.e.， the Camassa-Holm equation (denoted as (CH) equation ([1， 19])) (1.1.1) where u is the velocity of fluid in the direction x， or equivalently， the height from the horizontal bottom to the free surface of the water wave， ω is a con-stant associated with the critical shallow wave velocity， and the subscripts ex-press partial derivatives. The equation contains the higher-order term (right branch) in small-amplitude expansion of the incompressible Euler equation with undirected motion on the free-surface water waves under the influence of gravity. The Benjamin-Bona-Mahoney (BBM) equation can be derived by removing these terms as follows: or the Korteweg-de Vries (KdV) equation (1.1.1) as an expanded form of the BBM equation. And the first derivative of the soliton solution c exp(|x-ct|) of the limit form is discontinuous at the crest when ω ≈ 0， which is called the peakon solution. These peakons affect the solutions of the initial value problem of Equation (1.1.1) as ω =0. The way in which the smooth initial conditions evolve into a series of peakons during the change is that the first derivative appears discontinuously by changing each inflection point with a negative slope to have a vertical tangent line. It should be noted that we can get multiple soliton solutions by simply overlapping single peakon solutions in a completely integrable finite dimensional Hamiltonian system and calculating their amplitude variations and the positions of the peaks. The CH equation has a double Hamiltonian structure， which can be expressed in two different Hamiltonian forms. The ratio of two compatible Hamiltonian operators is a recursive operator， resulting in an infinite number of conserved quantities. The property of the double Hamiltonian is used to rewrite our equa-tion as a compatibility condition of linear isospectral problems. So the initial value problem can be solved by the inverse scattering transform (IST). The Undirected Model Consider the inviscous incompressible fluid equa-tion with the same density: the Euler equation. Here u represents the horizontal speed in the x direction， and w represents the speed value in the z axis (vertical direction). The fluid is moved affected by gravity g， which is on a free surface z = ζ(x， t) and the horizontal infinite domain of the flat bottom z = .h0. We substitute the solutions formed asymptotically by the shallow water wave (see [1])， i.e.， u = u(x， t) and w =-(z + h0)ux into the conserved quantity (power + potential) of the Euler equation. Then we can obtain the energy by integrating z as follows: where η = ζ +h0 is the height from the water wave to the bottom. In addition， we substitute the same solution into the Euler equation and integrate in the vertical coordinate to obtain the Green-Naghdi (GN) equation ([2]). The GN equation is conserved about the above energy HGN. In fact， they can be expressed in the form of Hamiltonian ([3]) as follows: (1.1.2) where the momentum density m is defined as m = δHGN/δu. The GN equation does not allow for a thin domain expansion over the small parameter ε which rep-resents the proportion between the depth and the wavelength. In this expansion， the momentum of movement of HGN in the vertical direction (~η3ux2 ) is O(ε2). We make further small amplitude assumptions about shallow-wave theory， which is in the form of η = h0 + O(α)， and with balance quantity α = O(ε2). Conversely， the Hamiltonian HGN keeps the higher order non-essential terms (like ζ3) in such expansion. From the GN equation， we make further asymp
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