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 11.1 The exact solutions in the invariant manifold of the saddle-saddle 228 11.2 Exact solitary wave solution and quasi-periodic solutions in the invariant manifold of the saddle-center 233 11.3 Exact periodic wave solutions and quasi-periodic wave solutions in the invariant manifold of the center-center 237 Chapter 12 Exact Solutions in the Invariant Manifolds of the Generalized Integrable H.enon-Heiles System and Exact Traveling Wave Solutions of Klein-Gordon-Schr.odinger Equations 240 12.1 Exact solutions in the invariant manifolds MK1K2 under the integrable condition (i) 242 12.2 Exact solutions in the invariant manifold of the equilibrium point E0 under the integrable condition (ii) 253 12.3 Exact solutions in the invariant manifolds MK1K2 under the integrable condition (iii) 260 12.4 Exact traveling wave solutions of Klein-Gordon-Schr.odinger equations 265 Chapter 13 Exact Solutions and Bifurcations in Invariant Manifolds for A Nonic Derivative Nonlinear Schr.odinger Equation 267 13.1 Bifurcations of phase portraits of system (13.12) in two cases of quintic and nonic nonlinearities of equation (13.1) 270 13.2 The exact solutions of equation (13.1) with quintic nonlinearity given from system (13.14) with q = 2 273 13.3 The exact solutions of equation (13.1) with nonic nonlinearity given from system (13.14) with q = 3 283 13.4 Bifurcations of phase portraits of system (13.12) when f(.) has four positive zeros 300 Chapter 14 Exact Homoclinic Orbits and Heteroclinic Families for A Third-Order System in the Chazy Class XI (N = 3) 303 14.1 Exact homoclinic orbits and heteroclinic cycle family of system (14.2) in the level surface .(x1; x2; x3) = K1 306 14.2 Uncountably inˉnitely many periodic solutions and unbounded solutions of system (14.2) in the level set .(x1; x2; x3) = Kc; Kc 2 [.K1;K1) 313 14.3 Uncountably inˉnitely many unbounded solutions of (14.2) in the level set .(x1; x2; x3) = Ks; Ks 2 (.1;.K1) and Ks 2 (K1;1) 317 14.4 Chaotic dynamics of perturbed systems of (14.2): numerical examples 319 Chapter 15 Exact Heteroclinic Cycle Family and Quasi-Periodic Solutions for the Three-Dimensional System Determined by Chazy Class IX 321 15.1 Two heteroclinic orbit families in the level surface (x1; x2; x3) = Ke 323 15.2 Exact quasi-periodic solutions and periodic solutions 330 Bibliography 332
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