本书是一本用英文写成的数学类教材，是作者基于多年的科研和全英文教学经验编写而成的。全书分为10章。前3章是预备知识和方法，包含了某些数学软件程序、某些函数和积分公式以及平面系统的相图等内容。后7章是针对7个著名方程所描述的非线性波进行数值模拟和推导其表达式，包含KdV方程的行波、mKdVI方程的孤立波和周期波、mKdVⅦ方程的扭波与周期波、Gardner方程的三角周期波及其极限、Camassa-Holm方程的孤立波与周期尖波、特殊广义b-方程的反孤立波分支、广义Camassa-Holm方程的孤立尖波分支。本书力求做到讲解由浅入深、数值模拟与理论推导相结合、图文并茂、通俗易懂。
本书可作为高等院校数学类、物理类、力学类等专业的高年级本科生或研究生的微分方程后续课程教材，也可作为对微分方程的相平面分析和数值模拟感兴趣的朋友的自学读本。

Chapter 1 Some Codes of the Software Mathematica
Exercise
Chapter 2 Some Functions and Integral Formulas
2.1 Hyperbolic Functions
2.2 Elliptic Sine and Cosine Functions
2.3 Some Integral Formulas
Exercise
Chapter 3 Phase Portraits of Planar Systems
3.1 Standard Forms of Linear Systems
3.2 Classification of Singular Points for Linear Systems
3.3 Phase Portraits and Their Simulation for Some Linear Systems
3.4 Properties of Singular Points of Nonlinear Systems with Nonzero Eigenvalues
3.5 The Standard Forms of Nonlinear Systems with Zero Eigenvalues
3.6 Properties of Singular Points of Systems with Zero Eigenvalues
Exercise
Chapter 4 The Traveling Wave of KdV Equation
4.1 The Phase Portrait of System (4.7)
4.2 The Solitary Wave Solution
4.3 Elliptic Sine Smooth Wave Solution
4.4 Limit of Elliptic Sine Smooth Wave Solution
4.5 Hyperbolic Blow-up Wave Solution
4.6 Trigonometric Blow-up Wave Solution
4.7 Elliptic Sine Blow-up Wave Solution
4.8 Elliptic Cosine Blow-up Wave Solution
4.9 Fractional Blow-up Wave Solution
Exercise
Chapter 5 The Solitary Wave and Periodic Wave of mKdVI Equation
5.1 Phase Portrait of System (5.7)
5.2 Hyperbolic Solitary Wave Solution
5.3 Elliptic Sine Smooth Wave Solution and Their Limits
5.4 Elliptic Cosine Smooth Wave Solution and Their Limits
5.5 Trigonometric Smooth Periodic Wave Solution
5.6 Fractional Solitary Wave Solution
Exercise
Chapter 6 The Kink Wave and Periodic Wave of mKdVII Equation
6.1 Phase Portrait of System (6.7)
6.2 Kink Wave Solution
6.3 Smooth Periodic Wave Solution
6.4 Elliptic Cosine Blow-up Wave Solution and Their Limits
6.5 Elliptic Sine Blow-up Wave Solution
Exercise
Chapter 7 The Trigonometric Smooth Periodic Wave Solutions and Their Limits of Gardner Equation
7.1 Singular Points and Their Properties
7.2 Bifurcations Lines
7.3 The Roots of H(φ, 0) = hi
7.4 Bifurcation Phase Portraits
7.5 The Expressions of Trigonometric Smooth Periodic Wave Solutions and Their Limits
7.6 The Derivations for the Expressions of the Trigonometric Periodic Wave Solutions and Their Limit Forms
Exercise
Chapter 8 The Peakon and Periodic Cusp Wave of Camassa-Holm Equation
8.1 The Traveling Wave System and Its Accompany System
8.2 The Distributions of Singular Points for System (8.10)
8.3 The Properties of the Singular Points for System (8.10)
8.4 The Values of H(φ, y) at the Singular Points and the Graphs of H(φ, y) = h
8.5 The Single-Soliton and Peakon of Eq.(8.1)
8.6 The Peakon Solution
8.7 The Periodic Cusp Wave
Exercise
Chapter 9 The Double Bifurcation of Anti-Solitary Waves in the Special Genralized b-Equation
9.1 The Traveling Wave System and Its Accompany System
9.2 The First Integration of Systems (9.14) and (9.18)
9.3 The Distributions of Singular Points of System (9.18)
9.4 The Properties of the Singular Points System (9.18)
9.5 The Bifurcation Phase Portraits of System (9.18)
9.6 The Bifurcation of the Anti-Solitary Waves of Eq.(9.1)
9.7 The Expressions and Bifurcations of the Anti-Solitary Waves of Eq.(9.1)
9.8 The Bifurcations of An Anti-Solitary Wave
Exercise
Chapter 10 The Bifurcations of Peakons in a Generalized Comassa-Holm Equation
10.1 Traveling Wave System and Its Bifurcation Phase Portraits
10.2 The Hyperbolic Peakon Wave Solutions
10.3 The Fractional Peakon Wave Solutions
10.4 The Bifurcations of Peakon Wave Solutions
Exercise
References