Studies On Exact Traveling Wave Solutions For Some Nonlinear Wave Equations

In this thesis,by using dynamical system method and singular traveling wave equa-tion theory,the exact traveling wave solutions of several kinds of nonlinear wave equations which have physical significance are studied.These equations include the generalized two-component peakon-type dual equations,the rotation-Camassa-Holm equation,a nonlocal hydrodynamic-type equation and the fractional mKdV equation.In the thesis,the dy-namic properties of the singular traveling wave systems corresponding to these nonlinear wave equations are analyzed in detail,as well as the bifurcation behavior of the systems which varies with the parameters.Using tools such as elliptic function,abundant exact traveling wave solutions of the systems are obtained through complex calculation.The thesis is divided into seven chapters,and the major works are as follows:In Chapter 1,we introduce the history of the development of soliton theory.Several important methods of solving nonlinear wave equation are introduced.Also,the main research content and results are illustrated.In Chapter 2,we introduce some basic knowledge related to this thesis,including dy-namical system and differential equation,dynamical system method for singular nonlinear wave equation.In Chapter 3,we study the bifurcation and exact traveling wave solutions of two gen-eralized two-component peakon-type dual equations.One of them contains the famous two-component Camassa-Holm equation.Using dynamical system method and singular traveling wave equation theory,the two equations are reduced to the same planar dy-namical system.By qualitative analysis of the singular traveling wave system,the phase portraits of the corresponding traveling wave system is drawn,and as many exact trav-eling wave solutions as possible are obtained,including solitary wave solution,peakon,pseudo-peakon,periodic peakon,compacton,etc.Through comprehensive comparison and analysis,it is found that the distribution of these traveling wave solutions follows certain rules.In Chapter 4,we study the bifurcation and exact traveling wave solutions of the rotation-Camassa-Holm equation which contains the famous Camassa-Holm equation and is a special case of the generalized Camassa-Holm equation.By using dynamical system method and singular traveling wave equation theory,the bifurcations of phase portraits with different parameters in a parameter space with five parameters are studied.Smooth solitary wave solution,periodic wave solution,peakon,periodic peakon as well as com-pacton and their exact representations are obtained.Moreover,from each group of phase portraits,it can be clearly seen that the singular straight line has a great influence on the changes of phase portraits and the generation of bifurcations.In Chapter 5,the bifurcation and exact traveling wave solution of a nonlocal hy-drodynamic type equation is investigated.By dynamical system method and singular traveling wave equation theory,all kinds of exact traveling wave solutions,including s-mooth solitary wave solution,the uncountably infinitely many solitary wave solutions,pseudo-peakon,periodic peakon,compacton,kink and anti-kink wave solution and so on are derived.In particular,the uncountably infinitely many solitary wave solutions,kink and anti-kink wave solutions are new.Also,the uncountably infinitely many solitary wave solutions are different from the general smooth solitary wave solution.The uncountably infinitely many homoclinic orbits at the higher order equilibrium point correspond to the uncountably infinitely many solitary wave solutions.It is a very peculiar phenomenon.In Chapter 6,we study the bifurcation and exact traveling wave solutions of the mKdV equation with conformable fractional derivative.The fractional partial differen-tial equation is reduced to an ordinary differential equation dependent on the fractional order ? by traveling wave transformation.Then,the dynamical system method is used to analyze the phase portraits of the corresponding traveling wave system,and the exact traveling wave solutions of the original system are obtained,including the smooth soli-tary wave solution,periodic wave solution,kink and anti-kink wave solution.Through analysis,it is found that the solution of fractional mKdV equation has the basic form of the solution of general mKdV equation,and its wave width and amplitude depend on the fractional order ?.In Chapter 7,we summarize the main work in this thesis and give some problems for further exploration.