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#### Phase Plane Analysis and Numerical Simulation of Wae Equations

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Contents
Chapter 1 Some Codes of the Software Mathematica 1
Exercise 15
Chapter 2 Some Functions and Integral Formulas 17
2.1 Hyperbolic Functions 17
2.2 Elliptic Sine and Cosine Functions 18
2.3 Some Integral Formulas 21
Exercise 24
Chapter 3 Phase Portraits of Planar Systems 25
3.1 Standard Forms of Linear Systems 25
3.2 Classification of Singular Points for Linear Systems 28
3.3 Phase Portraits and Their Simulation for Some Linear Systems 32
3.4 Properties of Singular Points of Nonlinear Systems with Nonzero
Eigenvalues 40
3.5 The Standard Forms of Nonlinear Systems with Zero Eigenvalues 50
3.6 Properties of Singular Points of Systems with Zero Eigenvalues 52
Exercise 55
Chapter 4 The Traveling Wave of KdV Equation 56
4.1 The Phase Portrait of System (4.7) 56
4.2 The Solitary Wave Solution 61
4.3 Elliptic Sine Smooth Wave Solution 64
4.4 Limit of Elliptic Sine Smooth Wave Solution 67
4.5 Hyperbolic Blow-up Wave Solution 68
4.6 Trigonometric Blow-up Wave Solution 70
4.7 Elliptic Sine Blow-up Wave Solution 71
4.8 Elliptic Cosine Blow-up Wave Solution 74
4.9 Fractional Blow-up Wave Solution 77
Exercise 80
Chapter 5 The Solitary Wave and Periodic Wave of mKdVI Equation 82
5.1 Phase Portrait of System (5.7) 82
5.2 Hyperbolic Solitary Wave Solution 86
5.3 Elliptic Sine Smooth Wave Solution and Their Limits 89
5.4 Elliptic Cosine Smooth Wave Solution and Their Limits 93
5.5 Trigonometric Smooth Periodic Wave Solution 97
5.6 Fractional Solitary Wave Solution 101
Exercise 103
Chapter 6 The Kink Wave and Periodic Wave of mKdVII Equation 104
6.1 Phase Portrait of System (6.7) 105
6.2 Kink Wave Solution 110
6.3 Smooth Periodic Wave Solution 113
6.4 Elliptic Cosine Blow-up Wave Solution and Their Limits 115
6.5 Elliptic Sine Blow-up Wave Solution 116
Exercise 118
Chapter 7 The Trigonometric Smooth Periodic Wave Solutions and Their Limits of Gardner Equation 120
7.1 Singular Points and Their Properties 120
7.2 Bifurcations Lines 121
7.3 The Roots of H(φ， 0) = hi 122
7.4 Bifurcation Phase Portraits 123
7.5 The Expressions of Trigonometric Smooth Periodic Wave Solutions and Their Limits 125
7.6 The Derivations for the Expressions of the Trigonometric Periodic Wave Solutions and Their Limit Forms 127
Exercise 135
Chapter 8 The Peakon and Periodic Cusp Wave of Camassa-Holm Equation 137
8.1 The Traveling Wave System and Its Accompany System 137
8.2 The Distributions of Singular Points for System (8.10) 139
8.3 The Properties of the Singular Points for System (8.10) 140
8.4 The Values of H(φ， y) at the Singular Points and the Graphs of H(φ， y) = h 145
8.5 The Single-Soliton and Peakon of Eq.(8.1) 151
8.6 The Peakon Solution 155
8.7 The Periodic Cusp Wave 159
Exercise 163
Chapter 9 The Double Bifurcation of Anti-Solitary Waves in the Special Genralized b-Equation 165
9.1 The Traveling Wave System and Its Accompany System 165
9.2 The First Integration of Systems (9.14) and (9.18) 167
9.3 The Distributions of Singular Points of System (9.18) 168
9.4 The Properties of the Singular Points System (9.18) 171
9.5 The Bifurcation Phase Portraits of System (9.18) 177
9.6 The Bifurcation of the Anti-Solitary Waves of Eq.(9.1) 177
9.7 The Expressions and Bifurcations of the Anti-Solitary Waves of Eq.(9.1) 181
9.8 The Bifurcations of An Anti-Solitary Wave 183
Exercise 185
Chapter 10 The Bifurcations of Peakons in a Generalized Comassa-Holm Equation 188
10.1 Traveling Wave System and Its Bifurcation Phase Portraits 188
10.2 The Hyperbolic Peakon Wave Solutions 196
10.3 The Fractional Peakon Wave Solutions 199
10.4 The Bifurcations of Peakon Wave Solutions 201
Exercise 207
References 208
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