Dynamical Behavior And Exact Solutions Of Nonlinear Traveling Wave Equations

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 real-world 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 the 19th 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,Schrodinger equation,Einstein equation,Yang-Mills equation,and reaction-diffusion equation,etc.More and more partial differential equations,especially the nonlinear equa-tions 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 non-linear mathematical physics,especially in the studies of soliton theory.The pri-mary 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 singu-lar 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 de-scribed above and stated as follows.What is the dynamical behavior of exact traveling wave solutions?How do traveling wave solutions depend on the parame-ters 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 Hamiltonian structure.In Chapter four and five,we considered the model of derivative of nonlinear Schrodingers equation of septic and thirteenth order under invariants in a resonant nonlinear model.It is shown that a wave packet ansatz inserted into these equa-tions 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 Schrodingers equa-tion.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 be-havior.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 Naranman-dula 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 eas-ily 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 so-lutions 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 corre-sponding 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 prop-erties 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,ex-istence 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.