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。本书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。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。

Chapter 1 Physical Backgrounds and Complete Integrability of the Camassa-Holm Equation
1.1 Physical backgrounds of the Camassa-Holm equation
1.2 The complete integrability of the Camassa-Holm equation
1.3 Experimental observation and applications of solitons
References
Chapter 2 Traveling Wave Solutions of the Camassa-Holm Equation
2.1 Introduction
2.2 Notations
2.3 Weak form
2.4 Several types of traveling wave solutions
2.6 The correlation of parameters
2.7 Wave length
2.8 Explicit formulae of peakon
References
Chapter 3 The Scattering and Inverse Scattering of the Camassa-Holm Equation
3.1 Scattering of the Camassa-Holm equation
3.1.1 Introduction
3.1.2 Spectral graph theory
3.1.3 The scattering problem
3.2 The solutions of Camassa-Holm equation
3.2.1 Introduction
3.2.2 Summary of the process
3.2.3 Summary of solving process
3.2.4 Solitary wave solutions
3.2.5 2-soliton solutions
3.2.6 Examples and properties of 2-soliton solutions
3.2.7 3-soliton solutions
3.2.8 Summary
References
Chapter 4 Well-posedness of the Camassa-Holm Equation
4.1 Global existence of strong solutions
4.1.1 The existence of local solutions
4.1.2 The existence of global solutions
4.2 The existence of global weak solutions
4.3 The local well posedness of the Cauchy problem to the Camassa-Holm equation in Hs (s>3-2)
4.4 Blow-up phenomena of the Camassa-Holm equation
4.5 The orbital stability of peakon solutions
References
Chapter 5 Formation and Dynamics Analysis of Shock Wave of the Degasperis-Procesi Equation
5.1 Introduction
5.1.1 Peakons
5.1.2 Generalized weak solutions
5.2 The shock wave of the DP equation
5.3 Peakon, anti-peakon and the formation of shock waves
5.4 Shock wave dynamics
5.5 Concluding remarks
References
Chapter 6 Water Wave Structure and Nonlinear Equilibrium of b-Family Nonlinear Shallow Water Wave Equation
6.1 Introduction
6.1.1 b-family shallow water wave equation
6.1.2 Outline of the paper
6.2 History and general properties of the b-equation
6.2.1 Discrete symmetries: reversibility, parity, and signature..
6.2.2 Lagrangian representation
6.2.3 Preservation of the norm
6.2.4 Lagrangian representation for integer b
6.2.5 Reversibility and Galilean covariance
6.2.6 Integral momentum conservation
6.3 Traveling waves and generalized functions
6.3.1 The case of b=0
6.3.2 The case of b≠0
6.3.3 The case orb>0
6.3.4 The case orb<0
6.4 Pulson interactions for b>0
6.4.1 Pulson interactions for b＝2
6.4.2 Peakon interactions for b＝2 and b=3: numerical results
6.4.3 Pulson-pulson interactions for b>0 and symmetric g
6.4.4 Pulson-antipulson interactions for b>1 and symmetric g
6.4.5 Specializing pulsons to peakons for b = 2 and b = 3
6.5 Peakons of width a for arbitrary b
6.5.1 The slope dynamics of peakons: inflection points and the steepening lemma when 16.5.2 The case of 0≤ 6.6 Adding viscosity to peakon dynamics
6.6.1 Burgers-αβequation: analytical estimates
6.6.2 The traveling waves of Burgers-αβ equation forβ(3- b) = 1 and v = 0
6.7 The peakons under (6.1.1 ) adding viscosity and evolution of(6.1.2 ) Burgers-αβ
6.7.1 The fate of peakons under adding viscosity
6.7.2 The fate of peakons under Burgers-αβ evolution
6.8 Numerical results for peakon scattering and initial value problems
6.8.1 Peakon initial value problems
6.8.2 Description of our numerical methods
6.9 Conclusions
References
Chapter 7 The Degasperis-Procesi Equation
7.1 Introduction
7.2 Local well-posedness
7.3 Blow up
7.4 Global strong solutions
7.5 Global weak solutions
7.6 Recent results and problems
References