Singular Nonlinear Travelling Wave Equations:Bifurcations and Exact Solutions

  The aim of this book is to give a more systematic account for the bifurcation theory method of dynamical systems to find exact travelling wave solutions and their dynamics with an emphasis on the understanding of the above mentioned "new wave" for two classes of singular nonlinear travelling equations 

Chapter 1 Some Physical Models Which Yield Two Classes of Singular Travelling Wave Systems
1.1 Nonlinear wave equations having the first class of singular nonlinear travelling wave systems
1.2 Nonlinear wave equations having the second class of singular nonlinear travelling wave equations
Chapter 2 Dynamics of Solutions of Singular Travelling Systems
2.1 Some preliminary knowledge of dynamical systems
2.2 Phase portraits of travelling wave systems having singular straight lines
2.3 Dynamical behavior of orbits in neighborhoods of the singular
straight line: the case of $1,2 are saddle points
2.4 Dynamical behavior of orbits in neighborhoods of the singular straight line: the case of $1,2 are node points
2.5 Dynamical behavior of orbits divided by the singular curves:
singular travelling wave systems of the second class
Chapter 3 Exact Travelling Wave Solutions and Their Bifurcations for the Kudryashov-Sinelshchikov Equation
3.1 Bifurcations of phase portraits of system (3.0.4)
3.2 Exact travelling wave solutions for/ -- -3,-4
3.3 Exact travelling wave solutions for/ --- 1, 2
Chapter 4 Bifurcations of Travelling Wave Solutions of Generalized Camassa-Holm Equation (I)
4.1 Bifurcations of phase portraits of (4.0.2)
4.2 The exact parametric representations of travelling wave solutions of (4.0.1)
4.3 The existence of smooth solitary wave solutions and periodic wave solutions
Chapter 5 Bifurcations of Travelling Wave Solutions of Higher Order Korteweg-De Vries Equations
5.1 Travelling wave solutions of the second order Korteweg-De Vries equation in the parameter condition group (I)
5.2 Travelling wave solutions of the second order Korteweg-De Vries
equation in the parameter condition group (II)
5.3 Travelling wave solutions for the generalization form of modified Korteweg-De Vries equation
Chapter 6 The Bifurcations of the Travelling Wave Solutions of K(m, n) Equation
6.1 Bifurcations of phase portraits of system (6.0.2)
6.2 Some exact explicit parametric representations of travelling wave solutions
6.3 Existence of smooth and non-smooth solitary wave and periodic wave solutions
6.4 The existence of uncountably infinite many breaking wave
solutions and convergence of smooth and non-smooth travelling
wave solutions as parameters are varied
Chapter 7 Kink Wave Solutions Determined by a Parabola Solution of Planar Dynamical Systems
7.1 Six classes of nonlinear wave equations
7.2 Existence of parabola solutions of (7.1.2) and their parametric representations
7.3 Kink wave solutions for 6 classes of nonlinear wave equations
Chapter 8 Exact Dark Soliton, Periodic Solutions and Chaotic Dynamics
in a Perturbed Generalized Nonlinear Schrhdinger Equation
8.1 The exact solutions of (8.0.2) for the cubic NLS equation with f(q) = aq
8.2 The exact solutions of (8.0.2) for the cubic-quintic NLS equation with f(q) = cq q2
8.3 The persistence of dark solition for the perturbed cubic-quintic NLS equation (8.0.12) without the term V(x)u
8.4 Chaotic behavior of the travelling wave solutions for the perturbed cubic-quintic NLS equation (8.0.12)
Chapter 9 Bifurcations and Some Exact Travelling Wave Solutions of a Generalized Camassa-Holm Equation (II)
9.1 Bifurcations of phase portraits of (9.0.5)
9.2 Some exact travelling wave solution of (9.0.2) in the symmetry cases.
9.3 The exact travelling wave solutions of equation (9.0.2) in a non-symmetric case
Chapter 10 Bifurcations of Breather Solutions of Some Nonlinear Wave Equations
10.1 Introduction
10.2 Bifurcations of travelling wave solutions of system (10.1.7) when VRp(0, r) given by (10.1.2)
10.3 Travelling wave solutions of system (10.1.1) with Vap(0, r) given by (10.1.2)
10.4 Bifurcations of solutions of (10.1.7) with VRp(0, r) given by (10.1.3).
10.5 Travelling wave solutions of (10.1.1) with VRP(O, r) given by (10.1.3)
10.6 Bifurcations of breather solutions of (10.1.4)
Chapter 11 Bounded Solutions of (n + 1)-dimensional Sine-and Sinh-Gordon Equations
11.1 (n + 1)-dimensional Sine-and Sinh-Gordon equations
11.2 The bounded solutions of the systems (11.1.4) and (11.1.5)
11.3 The bounded travelling wave solutions of the form (ll.l.2a) of (ll.l.la)
Chapter 12 Exact Loop Solutions and Their Dynamics of Some Nonlinear Wave Equations
12.1 The elastic beam equation
12.2 The reduced Ostrovsky equation
12.3 The short pulse equation
12.4 More nonlinear wave equations which have breaking loop-solutions-.
Chapter 13 Exact Solitary Wave, Periodic and Quasi-periodic Wave Solutions for the KdV6 Equations
13.1 The equilibrium points and linearized systems of (13.0.6)
13.2 Exact solitary wave and quasi-periodic wave solutions of the CDG equation (13.0.12)
13.3 Exact solutions of the Kaup-Kupershmidt equation (13.0.10)
13.4 Exact solutions of the KdV6 equation (13.0.2)
13.5 Exact solutions of the KdV6 equation (13.0.3)
Chapter 14 Exact Travelling Wave Solutions and Their Dynamics for a Class Coupled Nonlinear Wave Equations
14.1 Exact explicit solutions y = xl() of (14.0.4a) when P(t) has the factorization (14.1.1)
14.2 Some properties of solutions v() of equation (14.0.45)
Chapter 15 On the Travelling Wave Solutions for a Nonlinear Diffusion-convection Equation
15.1 The dynamics of the travelling wave solutions and the existence of global monotonic wavefront solutions of (15.0.1)
15.2 Dynamical behavior of system (15.0.3)
15.3 Exact travelling wave solutions of (15.0.3)
References