《奇非线性波方程——分支和精确解》由李继彬编著。

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.

《奇非线性波方程——分支和精确解》由李继彬编著。

《奇非线性波方程——分支和精确解》讲述了：

The studies of solitons and complete integrability of nonlinear wave equations and

   bifurcations and chaos of dynamical systems are two very active fields in nonlinear science. Because a homoclinic orbit of a travelling wave system (ODEs) corresponds to a solitary wave solution of a nonlinear wave equation (PDE). This fact provides an intersection point for the above two study fields. 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 singular properties (so called "new waves mathematics") of solutions, such as peakons, cuspons, compactons and loop solutions et al., for some classes of very well known nonlinear wave equations. Readers shall find how standard methods of the theory of dynamical systems may be used for the study of travelling wave solutions even in the case of systems with discontinuities.

Any reader trying to understand the subject of this book is only required to know the elementary theory of dynamical systems and elementary knowledge of nonlinear wave equations. This book should be useful as a research reference for graduate students, teachers and engineers in different study fields.

Preface

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