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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。
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ContentsChapter 1 Physical Backgrounds and Complete Integrability of the Camassa-Holm Equation 11.1 Physical backgrounds of the Camassa-Holm equation 11.2 The complete integrability of the Camassa-Holm equation 91.3 Experimental observation and applications of solitons 17References 18Chapter 2 Traveling Wave Solutions of the Camassa-Holm Equation 332.1 Introduction 332.2 Notations 342.3 Weak form 362.4 Several types of traveling wave solutions 372.5 The proof Theorem 2.4.1 432.6 The correlation of parameters 592.7 Wave length 632.8 Explicit formulae of peakon 66References 68Chapter 3 The Scattering and Inverse Scattering of the Camassa-Holm Equation 713.1 Scattering of the Camassa-Holm equation 713.1.1 Introduction 713.1.2 Spectral graph theory 723.1.3 The scattering problem 823.2 The solutions of Camassa-Holm equation 893.2.1 Introduction 893.2.2 Summary of the process 893.2.3 Summary of solving process 933.2.4 Solitary wave solutions 933.2.5 2-soliton solutions 973.2.6 Examples and properties of 2-soliton solutions 1023.2.7 3-soliton solutions 1063.2.8 Summary 109References 111Chapter 4 Well-posedness of the Camassa-Holm Equation 1134.1 Global existence of strong solutions 1134.1.1 The existence of local solutions 1134.1.2 The existence of global solutions 1204.2 The existence of global weak solutions 1284.3 The local well posedness of the Cauchy problem to the Camassa-Holm equation in Hs (s > 2/3) 1384.4 Blow-up phenomena of the Camassa-Holm equation 1454.5 The orbital stability of peakon solutions 153References 157Chapter 5 Formation and Dynamics Analysis of Shock Wave of the Degasperis-Procesi Equation 1595.1 Introduction 1595.1.1 Peakons 1595.1.2 Generalized weak solutions 1625.2 The shock wave of the DP equation 1645.3 Peakon， anti-peakon and the formation of shock waves 1725.4 Shock wave dynamics 1845.5 Concluding remarks 190References 191Chapter 6 Water Wave Structure and Nonlinear Equilibrium of 6-Family Nonlinear Shallow Water Wave Equation 1946.1 Introduction 1956.1.1 6-family shallow water wave equation 1956.1.2 Outline of the paper 1966.2 History and general properties of the 6-equation 1976.2.1 Discrete symmetries: reversibility， parity， and signature 1996.2.2 Lagrangian representation 1996.2.3 Preservation of the norm ||m||L1/b， 0≤b≤1 2006.2.4 Lagrangian representation for integer b 2016.2.5 Reversibility and Galilean covariance 2026.2.6 Integral momentum conservation 2026.3 Traveling waves and generalized functions 2036.3.1 The case of b = 0 2036.3.2 The case of b ≠ 0 2056.3.3 The case of b > 0 2066.3.4 The case of b < 0 2086.4 Pulson interactions for b > 0 2146.4.1 Pulson interactions for b = 2 2156.4.2 Peakon interactions for b = 2 and b = 3: numerical results 2156.4.3 Pulson-pulson interactions for b > 0 and symmetric g 2176.4.4 Pulson-antipulson interactions for b> 1 and symmetric g 2206.4.5 Specializing pulsons to peakons for b = 2 and b = 3 2226.5 Peakons of width a for arbitrary b 2236.5.1 The slope dynamics of peakons: inflection points and the steepening lemma when 1 6.5.2 The case of 0≤b≤1 2266.6 Adding viscosity to peakon dynamics 2266.6.1 Burgers-a^ equation: analytical estimates 2286.6.2 The traveling waves of Burgers-αβ equation for P(3 - b) = 1 and v = 0 2306.7 The peakons under (6.1.1) adding viscosity and evolution of (6.1.2) Burgers-αβ 2316.7.1 The fate of peakons under adding viscosity 2316.7.2 The fate of peakons under Burgers-ap evolution 2386.8 Numerical results for peakon scattering and initial value problems 2416.8.1 Peakon initial value problems 2416.8.2 Description of our numerical methods 2436.9 Conclusions 245References 247Chapter 7 The Degasperis-Procesi Equation 2497.1 Introduction 2497.2 Local well-posedness 2537.3 Blow up 2557.4 Global strong solutions 2597.5 Global weak solutions 2637.6 Recent results and problems 278References 284
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