The Studies of solitons and complete integrability of nonlinear wave equations and bifurcations and chaos of dynamical ystems are two very active fields in nonlinear science. Because a homoclinic orbit of a traveling wave system(ODEs) corresponds to a solitary wave solution of a nonlinear wave equation(PDE). This fact provides an intersection point for above two study fields. The aim of this book is to give a more systematic accotmt for the bifurcation theory method of dynamical systems to find traveling wave solutions with an emphasis on singular waves and understand their dynamics for some classes of the wellposedness of nonlinear evolution equations. Readers shall find how standard methods of the theory of dynamical systems may be used for the study of traveling wave solutions even 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.
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目录
Chapter 1 Traveling Wave Equations of Some PhysicalModels 1.1 The model of nonlinear oscillations of hyperelastic rods 1.2 Higher order wave equations of Korteweg-De Vries type 1.3 Camassa-Holm equation and its generalization forms 1.4 More classes of equations of mathematical physics Chapter 2 Basic Mathematical Theory of the Singular Traveling Wave Systems 2.1 Some preliminary knowledge of dynamical systems 2.2 Phase portraits of traveling wave equations having singular straight lines. 2.3 Main theorems to identify the profiles of waves and some examples Chapter 3 Bifurcations of Traveling Wave Solutions of Nonlinear Elastic Rod Systems 3.1 Bifurcations of phase portraits of (3.0.1) and physical acceptable solutions 3.2 Four types of solitary waves and three types of periodic waves 3.3 The non-periodic behavior of axial motions Chapter 4 Bifurcations of Traveling Wave Solutions of Generalized Camassa-Holm Equation 4.1 Bifurcations of phase portraits of system (4.0.2) 4.2 Exact parametric representations of traveling wave solutions of (4.0.1) 4.3 The existence of smooth solitary wave solutions and periodic wave solutions Chapter 5 Bifurcations of Traveling Wave Solutions of Higher Order Korteweg-De Vries Equations 5.1 Traveling wave solutions of the second order Korteweg-De Vries equation in the parameter condition group (I) 5.2 Traveling wave solutions of the second order Korteweg-De Vries equation in the parameter condition group (II) 5.3 Traveling wave solutions for the generalization form of modified Korteweg-De Vries equation Chapter 6 The Bifurcations of the Traveling 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 traveling 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 traveling wave solutions as parameters are varied Chapter 7 Kink Wave Solution Determined by a Parabola Solution of Planar Dynamical Systems 7.1 Six classes of nonlinear wave equations 7.2 Existence of parabola solutions and their parametric representations 7.3 Kink wave solutions of 6 classes of nonlinear wave equations Chapter 8 Traveling Wave Solutions of Coupled Nonlinear Wave Equations 8.1 Traveling wave equation of the Kupershmidt’s equation 8.2 Bifurcations of phase portraits of (8.1.7) 8.3 Existence of smooth solitary wave, kink wave and periodic wave solutions 8.4 Non-smooth periodic waves and uncountably infinite many breaking wave solutions Chapter 9 Solitary Waves and Chaotic Behavior for a Class of Coupled Field Equations 9.1 Solitary wave solutions of the integrable case of (9.0.1) 9.2 The existence of small amplitude periodic solutions 9.3 Chaotic behavior of solutions of (9.0.2) 9.4 The existence of arbitrarily many distinct periodic orbits Chapter 10 Bifurcations of Breather Solutions of Some Nonlinear Wave Equations 10.1 Introduction 10.2 Bifurcations of traveling wave solutions of system (10.1.7) when VRP (θ, r) given by (10.1.2) 10.3 Traveling wave solutions of system (10.1.1) with VRP (θ, r) givenby (10.1.2) 10.4 Bifurcations of solutions of (10.1.7) with VRP (θ, r) given by (10.1.3) 10.5 Traveling wave solutions of (10.1.1) with VRP (θ, r) given by (10.1.3) 10.6 Bifurcations of breather solutions of (10.1.4) Chapter 11 Bounded Solutions of (n+1)-Dimensional Sineand 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 traveling wave solutions of the form (11.1.2a) of(11.1.1) Chapter 12 Exact Explicit Traveling Wave Solutions for Two Classes of (n+1)-Dimensional Nonlinear Wave Equations 12.1 (n+1)-dimensional Klein-Gordon-Schrodinger equations 12.2 (n+1)-dimensional Klein-Gordon-Zakharov equations References
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