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　　内容简介

　　本书的**个目的是对行波解的分类和对奇异非线性行波方程所产生的峰、周期峰、伪峰和紧子的概念进行更系统的解释。从奇异摄动理论的动力系统和思想，我们证明周期性峰是行波系统的两个时间尺度光滑经典解。PeaKon是下限意义下的极限解：（i）在固定参数条件下，Peaon是一类周期性Peaon解的一个极限解；（ii）具有可变参数的Peaon是一个伪Pekon族的限制解。我们注意到，一个可积的非线性偏微分方程（非线性波动方程）的行波系统通常是一个可积的常微分方程组。因此，行波系统的相位轨道引起波函数的轮廓，并且行进系统的不同相位轨道引起波函数的不同轮廓。如果可能的话，这样的非线性行进系统，因为这些解析解对于理解波函数的性质是有用的。本书的第二个目的是引入动力系统方法寻找更具物理意义的可积系统的精确解。

　　内页插图

　　目录

　　Contents

　　Preface

　　Chapter 1 Some Shallow Water Wave Equations Which Yield Peakons and Compactons 1

　　1.1 Shallow water wave equations derived from the governing equations via double asymptotic power series expansions 1

　　1.2 Dynamics of traveling wave solutions to a new highly nonlinear shallow water wave equation 7

　　Chapter 2 Classiˉcation of Traveling Wave Solutions of the Singular Nonlinear Wave Equations 13

　　2.1 Some preliminary knowledge of dynamical systems 13

　　2.2 Bifurcations of phase portraits of travelling wave equations having singular straight lines 18

　　2.3 Main theorems to identify the wave proˉles for a singular traveling wave systems of the ˉrst class 23

　　2.4 Classiˉcation of the proˉles of traveling wave solutions via known phase orbits 28

　　Chapter 3 Exact Parametric Representations of the Orbits Deˉned by A Polynomial Di.erential Systems 54

　　3.1 Exact parametric representations of the orbits deˉned by the planar quadratic Hamiltonian systems 54

　　3.2 Exact parametric representations of the orbits deˉned by the symmetric planar cubic Hamiltonian systems 62

　　Chapter 4 Bifurcations and Exact Solutions of the Traveling Wave Systems for Dullin-Gottwald-Holm Equation 69

　　4.1 Bifurcations of phase portraits of systems (4.4) 70

　　4.2 Classiˉcation of all traveling wave solutions of system (4.4)+ and explicit exact parametric representations of the solutions of system (4:4)+ and (4.6) 72

　　4.3 Classiˉcation of all traveling wave solutions of system (4.4). and explicit exact parametric representations of the solutions of systems (4.6). and (4.4) 86

　　Chapter 5 Variform Exact One-Peakon Solutions for Some Singular Nonlinear Traveling Wave Equations of the First Kind 96

　　5.1 Peakon solutions of the generalized Camassa-Holm equation (5.1) 97

　　5.2 Peakon solutions of the nonlinear dispersion equation K(m; n) 101

　　5.3 Peakon solutions of the two-component Hunter-Saxton system (5.3) 104

　　5.4 Peakon solutions of the two-component Camassa-Holm system (5.4) 107

　　Chapter 6 Bifurcations and Exact Solutions of A Modulated Equation in A Discrete Nonlinear Electrical Transmission Line 111

　　6.1 Bifurcations of phase portraits of system (6.14) when f3(á) only has a positive zero 115

　　6.2 Dynamics and some exact parametric representations of the solutions of system (6.14) when f3(á) only has a positive zero 117

　　6.3 Bifurcations of phase portraits of system (6.14) when f3(á) has exact two positive zeros 123

　　6.4 Dynamics and some exact parametric representations of the solutions of system (6.14) when f3(á) has exact two positive zeros 126

　　Chapter 7 Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides， with Metamaterials， Having Polynomial Law Nonlinearity 129

　　7.1 Bifurcations of phase portraits of system (7.6) 132

　　7.2 Exact parametric representations of solutions of system (7.6) when there is only one equilibrium point for ˉ = 1; 2 and ˉ = .2;.3 136

　　7.3 Exact parametric representations of solutions of system (7.6) when there exist three equilibrium points for a0 > 0; ˉ = 1; 2 147

　　7.4 Exact parametric representations of solutions of system (7.6) when there exist three equilibrium points for a0 > 0; ˉ = .2;.3 163

　　7.5 Exact parametric representations of solutions of system (7.6) when there exist three equilibrium points for a0 < 0; ˉ = .3;.2; 1; 2 173

　　Chapter 8 Quadratic and Cubic Nonlinear Oscillators with Damping and Their Applications 178

　　8.1 Exact solutions and dynamics of the integrable quadratic oscillator with damping 179

　　8.2 Exact solutions and dynamics of the integrable cubic nonlinear oscillator with damping 184

　　8.3 Exact traveling wave solutions of the van der Waals normal form (8.1) and the Cha.ee-Infante equation (8.4) 188

　　Chapter 9 Dynamics of Solutions of Some Travelling Wave Systems Determined by Integrable Li.enard System 191

　　9.1 The ˉrst integrals of Li.enard equation (9.4) under Chiellini's integrability condition 192

　　9.2 Dynamics of travelling wave solutions of a integrable generalized damped sine-Gordon equation (9.7) 194

　　9.3 Dynamics of travelling wave solutions of the integrable Burgers equation with one-side potential interaction (9.8) 199

　　Chapter 10 Bifurcations and Exact Solutions in A Model of Hydrogen-Bonded-Chains 204

　　10.1 Bifurcations of phase portraits of system (10.2) 206

　　10.2 The parametric representations of some orbits deˉned by system (10.2) for > 0; ˉp0 6= 0 208

　　10.3 The parametric representations of some orbits deˉned by system (10.2) for < 0; ˉp0 6= 0 213

　　10.4 The parametric representations of some orbits deˉned by system (10.2) for ˉp0 = 0 or ˉ < 0 217

　　10.5 The parametric representations of some orbits intersecting transversely the singular straight line p = §p0 220

　　Chapter 11 Exact Solutions in Invariant Manifolds of Some Higher-Order Models Describing Nonlinear Waves 224