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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Deg***eris-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Deg***eris-Procesi方程的适定性给出具体证明。

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 applicati*** of solit*** 17

References 18

Chapter 2 Traveling Wave Soluti*** of the Camassa-Holm Equation 33

2.1 Introduction 33

2.2 Notati*** 34

2.3 Weak form 36

2.4 Several types of traveling wave soluti*** 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 soluti*** 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 soluti*** 93

3.2.5 2-soliton soluti*** 97

3.2.6 Examples and properties of 2-soliton soluti*** 102

3.2.7 3-soliton soluti*** 106

3.2.8 Summary 109

References 111

Chapter 4 Well-posedness of the Camassa-Holm Equation 113

4.1 Global existence of strong soluti*** 113

4.1.1 The existence of local soluti*** 113

4.1.2 The existence of global soluti*** 120

4.2 The existence of global weak soluti*** 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 soluti*** 153

References 157

Chapter 5 Formation and Dynamics Analysis of Shock Wave of the Deg***eris-Procesi Equation 159

5.1 Introduction 159

5.1.1 Peak*** 159

5.1.2 Generalized weak soluti*** 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 wate***ave 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 c***ervation 202

6.3 Traveling waves and generalized functi*** 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

*** Pulson in***cti*** for b > 0 214

***.1 Pulson in***cti*** for b = 2 215

***.2 Peakon in***cti*** for b = 2 and b = 3: numerical results 215

***.3 Pulson-pulson in***cti*** for b > 0 and symmetric g 217

***.4 Pulson-antipulson in***cti*** for b> 1 and symmetric g 220

***.5 Specializing puls*** to peak*** for b = 2 and b = 3 222

6.5 Peak*** of width a for arbitrary b 223

6.5.1 The slope dynamics of peak***: 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: ***ytical estimates 228

6.6.2 The traveling waves of Burgers-αβ equation for P(3 - b) = 1 and v = 0 230

6.7 The peak*** under (6.1.1) adding viscosity and evolution of (6.1.2) Burgers-αβ 231

6.7.1 The fate of peak*** under adding viscosity 231

6.7.2 The fate of peak*** 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 Deion of our numerical methods 243

6.9 Conclusi*** 245

References 247

Chapter 7 The Deg***eris-Procesi Equation 249

7.1 Introduction 249

7.2 Local well-posedness 253

7.3 Blow up 255

7.4 Global strong soluti*** 259

7.5 Global weak soluti*** 263

7.6 Recent results and problems 278

References 284

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书籍介绍

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，并且它是完全可积的,具有双哈密顿结构和Lax对。《Camassa-Holm方程》给出该类方程的物理背景并阐述它的完全可积性。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。《Camassa-Holm方程》同时研究含尖峰孤立子的Deg***eris-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Deg***eris-Procesi方程的适定性给出具体证明。