內容大鋼
本書是一本用英文寫成的數學類教材，是作者基於多年的科研和全英文教學經驗編寫而成的。全書分為10章。前3章是預備知識和方法，包含了某些數學軟體程序、某些函數和積分公式以及平面系統的相圖等內容。后7章是針對7個著名方程所描述的非線性波進行數值模擬和推導其表達式，包含KdV方程的行波、mKdVI方程的孤立波和周期波、mKdVⅦ方程的扭波與周期波、Gardner方程的三角周期波及其極限、Camassa-Holm方程的孤立波與周期尖波、特殊廣義b-方程的反孤立波分支、廣義Camassa-Holm方程的孤立尖波分支。本書力求做到講解由淺入深、數值模擬與理論推導相結合、圖文並茂、通俗易懂。
本書可作為高等院校數學類、物理類、力學類等專業的高年級本科生或研究生的微分方程後續課程教材，也可作為對微分方程的相平面分析和數值模擬感興趣的朋友的自學讀本。
目錄
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
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