Dynamical Behavior and Exact Solutions of Nonlinear Traveling Wave Equations

摘要

Nonlinear science is an interdisciplinary subject generated by the permeation of physics,mechanics,mathematics,and computer science,as well as the social sciences,notably in economics and demography.Most physical phenomena in realworld problems are described through nonlinear partial differential equations(are also called mathematical physics equations)involving first order or second order derivatives with respect to time,which play an important role to model a complex physical phenomenon in various fields.
　　From the19th century onwards a lot of new partial differential equations are derived,including the famous Laplace equation and Poisson equation are based on Newton's theory,wave equation,heat equation,Maxwell system,Schr(o)dinger equation,Einstein equation,Yang-Mills equation,and reaction-diffusion equation,etc.More and more partial differential equations,especially the nonlinear equations and systems will be derived owing to the progress of the modern science and technology.
　　In addition nonlinear wave equations are important mathematical models for describing natural phenomena and one of the forefront topics in the studies of nonlinear mathematical physics,especially in the studies of soliton theory.The primary objective of this study is to give more emphasis for the bifurcation method of dynamical systems to find traveling wave solutions with an emphasis on singular traveling wave equations and to understand the dynamics of some classes of nonlinear wave equations.
　　Basic question of this dissertation directly addressing the main objective described above and stated as follows.What is the dynamical behavior of exact traveling wave solutions?How do traveling wave solutions depend on the parameters of the dynamical system?What is the difference between singular systems and regular systems of traveling wave equations?Does existence of singular straight line in a dynamical system a necessary condition for the existence of nonsmooth dynamical behavior?This dissertation has four parts including eleven chapters.The major results and findings of this dissertation mainly are given below.
　　The first chapter is about introduction of nonlinear science,objective of the study,methodology used,basic questions of the study.Chapter two summarizes some qualitative research methods that are used to solve problems on nonlinear traveling wave equations.In this chapter the history,current research status and achievement of some methods,i.e.Hirota bilinear method,sub-equation methods,Lie symmetry method,and inverse scattering method are discussed.The last part of this chapter introduces the basic concepts on dynamical systems using the three-step method and their results developed and enriched by professor Jibin Li.
　　In Chapter three,by using dynamical system theory we construct the singular traveling wave system for a class of nonlinear evolution equations.These nonlinear wave equations exhibits a regular property and behavior,typically of integrable partial differential systems.In addition,we obtained a planar dynamical system and their traveling wave systems to study their HamiItonian structure.
　　In Chapter four and five,we considered the model of derivative of nonlinear Schr(o)dingers equation of septic and thirteenth order under invariants in a resonant nonlinear model.It is shown that a wave packet ansatz inserted into these equations leads to an integrable Hamiltonian dynamical sub-system.By studying the dynamical behavior and bifurcation of phase portraits of traveling wave system,we obtain exact parametric representations of traveling wave solutions.
　　In these two chapters we found interesting results,that is,firstly integrability of the model derived,a singular traveling wave systems and new regular system of the model was derived,which are new findings not determined before.Based on this results we determined new bifurcation curves for our study and depending on these level curves we obtained new physically important solutions.
　　Chapter six is devoted to the use of a combination of the wave transformation and the bifurcation method to generalized derivatives nonlinear Schr(o)dingers equation.Firstly,we have derived the dynamical system of the model and then we use them to reduce the equation to ordinary differential system.Under this study,even if there exists a singular line,the phase curves of the system are smooth(existence of singular straight line a sufficient condition for the appearance of a nonsmooth periodic orbit).
　　The orbits of the singular system which correspond to the homoclinic orbits of the regular system,for example,are still homoclinic orbits having smooth behavior.Then from the level curves,we obtained some exact solutions containing parameters,which are expressed by hyperbolic functions,trigonometric functions,and rational functions.In this chapter,we obtained new dynamical system and new important solutions such as periodic wave solutions,solitary wave solutions,and kink wave solutions.
　　Some studies in science may have incorrect results and researchers may use a different method to prove or disprove these results.In chapter seven we consider a model of a further modified Zakharov-Kuznetsov equation studied by Naranmandula and Wang to obtain new spiky and explosive solitary wave solutions by the use of a new simple function transformation.Here,we proved the existence of spiky and explosive solitary wave solutions,is not correct.Rather,by using parameter conditions we found different level curves and corresponding to each level curves we determined exact explicit periodic wave solutions,solitary wave solutions,and kink wave solutions.These findings pave a way to see a bifurcation theory is an exact one to find precisely corrected arguments.
　　In chapter eight,we found a new result on the Casimir equation for the Ito system by using the three-step method.After making time scale transformation,the singular traveling wave system is reduced to a regular system which is easily studied by classical bifurcation theory of dynamical systems and therefore the qualitative behavior of its orbits are obtained.Corresponding to the compacton solution,we obtained uncountably infinite many periodic wave solutions,bright and dark solitary wave solutions and kink wave solutions or anti-kink wave solutions determined.Corresponding to the positive or negative periodic solutions and homoclinic solutions of the derivative function system,there exist unbounded wave solutions of the wave function equation.
　　A generalized Dullin-Gottwald-Holm equation with the nonlinear dissipative term and nonlinear dispersive term are studied in Chapter nine.It is emphasized that the existence of singular straight line is the main cause for the appearance of non-smooth periodic cusp wave solutions and solitary cusp wave solutions.Various sufficient conditions to guarantee the existence of smooth and non-smooth traveling wave solutions are given.
　　The difference between singular system and the regular system is that(i)near the singular straight line,the periodic orbits have distinct time scales.(ii)in the area of the right side or the left side of the singular straight line,the direction of vector fields defined by the two systems are different.In this chapter,by taking advantage of singular perturbation theory,wonderful phenomena that the corresponding orbits between the singular system and the regular system have different dynamical behavior,are explained and strictly proved.These results obtained here enrich and develop the three-step method.
　　Chapter ten studies the qualitative behavior of traveling wave solutions of K(n,2n,-n)equation with negative exponents is studied.Since both their traveling wave systems have the singularity,after applying the qualitative theory of differential equations to studying the corresponding regular systems,the qualitative properties of all bounded orbits of the regular systems are obtained.Therefore the bifurcation parameter conditions which lead to smooth traveling wave solutions of K(n,2n,-n)equation are analyzed and the various sufficient conditions to guarantee the existence of the bounded traveling wave solutions are obtained.
　　For the K(n,2n,-n)equation with a negative exponent,the singularity does not cause the appearance of non-smooth traveling wave solutions,which shows that singular line does not always result in non-smooth solutions.That is to say,existence of singular straight line do not always mean for appearance of nonsmooth traveling wave solutions.The last chapter summarizes the main results of this study and future study directions.

目录

Abstract

Contents

List of Figures

Chapter 1 Background of the Study

1．1 Introduction

1．2 Objective，methodology，and basic research questions

Chapter 2 Some Qualitative Research Methods on Nonlinear Wave Equation

2．1 Highlights on some research methods of nonlinear wave equation

2．1．1 Hirota bilinear method

2．1．2 Sub-equation methods

2．1．3 Lie symmetry analysis method

2．1．4 Inverse scattering method

2．2 Basic concepts on dynamical system

2．2．1 Dynamical behavior of traveling wave systems

2．2．2 Two dimensional integrable Hamiltonian systems

2．2．3 Some properties Jacobi Elliptic function

Chapter 3 Studies on Singular Nonlinear Traveling Wave Systems：Dynamical Systems Approach

3．1 Models of higher order derivative nonlinear Schr(o)dinger equation

3．2 Zakharov-Kuznetsov equation and its further modified form

3．3 A generalized Dullin-Gottwald-Holm equation

3．4 Other models of mathematical physics considered

Chapter 4 Dynamical Behavior and Exact Solution inInvariant Manifold for a Septic Order Derivative Nonlinear Schr(o)dinger Equation

4．1 Introduction

4．2 Dynamical Behavior of system(4．1．1)when P=2

4．3 Exact parametric representations of traveling wave solutions(p：2)

4．4 Bifurcation of phase portraits of system(4．1．1)when P=3

4．5 Some traveling wave solutions of system(4．1．1)when P=3

Chapter 5 Exact Solution and Dynamical Behavior of Thirteenth Order Derivative Nonlinear Schr(o)dinger Equationl

5．1 Introduction

5．2 Dynamical Behavior of system(5．1．2)when β∈R

5．3 Parametric representation of exact solutions of system(5．1．2)β∈R

Chapter 6 Bifurcation and Exact Solution of a Generalized Derivative Nonlinear Schr(o)dingers Equation

6．1 Introduction

6．2 Bifurcations of phase portraits of system(6．1．2)γ∈R

6．3 Existence of smooth traveling wave solutions of system(6．1．2)γ∈R

Chapter 7 Exact Traveling Wave Solutions and Bifurcationsof a Further Modifled Zakharov-Kuznetsov Equation

7．1 Introduction

7．2 Bifurcations of phase portraits of system(7．1．2)when the origin is asaddle point

7．3 Parametric representations of exacttern(7．1．2)

7．3．1 Explicit parametric representations of the solutions of system(7．1．2)

7．3．2 More solutions applying the Fan sub-equation method for ZK-equation

Chapter 8 Existence of Kink and Unbounded Traveling Wave Solutions of the Casimir equation for the Ito System

8．1 Introduction

8．2 Bifurcations of phase portraits for of system(8．1．1)±

8．3 Exact explicit bounded traveling wave solutions of system(8．1．1)(±)

8．4 Existence of kink and unbounded traveling wave solutions

Chapter 9 Various Exact Solutions and Bifurcations of a Generalized Dullin-Gottwald-Holm Equation with a Power Law Nonlinearity

9．1 Introduction

9．2 Bifurcation of phase portraits of system(9．1．3)

9．3 Existence of smooth and non-smooth traveling wave solutions

9．4 Dynamical behavior of system(9．1．4)

9．5 Existence of smooth solitary，kink and periodic wave solutions

Chapter 10 Dynamical Behaviors of Traveling Wave Solutionsof a Generalized K(n，2n，-n)Equations

10．1 Introduction

10．2 Bifurcations of phase portraits of system(10．1．2)n=2，3

10．3 Traveling wave solutions of system(10．1．2)n=2

10．4 Some exact explicit parametric representation of system(10．1．2)n=3

Chapter 11 Main results and future direction

11．1 Main findings of the study

11．2 Future study direction

Bibliography

Publications

Acknowledgment

Curriculum Vitae

声明