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 < 0 208 6.4 Pulson interactions for 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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