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 Classi��cation 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 pro��les for a singular traveling wave systems of the ��rst class 23 2.4 Classi��cation of the pro��les of traveling wave solutions via known phase orbits 28 Chapter 3 Exact Parametric Representations of the Orbits De��ned by A Polynomial Di.erential Systems 54 3.1 Exact parametric representations of the orbits de��ned by the planar quadratic Hamiltonian systems 54 3.2 Exact parametric representations of the orbits de��ned 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 Classi��cation 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 Classi��cation 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(��) only has a positive zero 115 6.2 Dynamics and some exact parametric representations of the solutions of system (6.14) when f3(��) only has a positive zero 117 6.3 Bifurcations of phase portraits of system (6.14) when f3(��) has exact two positive zeros 123 6.4 Dynamics and some exact parametric representations of the solutions of system (6.14) when f3(��) 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 �� = 1; 2 and �� = .2;.3 136 7.3 Exact parametric representations of solutions of system (7.6) when there exist three equilibrium points for a0 > 0; �� = 1; 2 147 7.4 Exact parametric representations of solutions of system (7.6) when there exist three equilibrium points for a0 > 0; �� = .2;.3 163 7.5 Exact parametric representations of solutions of system (7.6) when there exist three equilibrium points for a0 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 ��rst 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 de��ned by system (10.2) for > 0; ��p0 6= 0 208 10.3 The parametric representations of some orbits de��ned by system (10.2) for 10.4 The parametric representations of some orbits de��ned by system (10.2) for ��p0 = 0 or �� 10.5 The parametric representations of some orbits intersecting transversely the singular straight line p = ��p0 220Chapter 11 Exact Solutions in Invariant Manifolds of Some Higher-Order Models Describing Nonlinear Waves 224