Research On Various Exact Traveling Wave Solutions Of Three Types Of Nonlinear Partial Differential Equation

In this paper,the exact traveling wave solutions of three types of nonlinear partial differential equations are solved by utilizing the method of bifurcations of dynamic systems.Firstly,we use the traveling wave transformations to convert the nonlinear partial differential equations to corresponding plane dynamic systems.Secondly,we obtain the bifurcation phase portraits under different parameters according to the qualitative theory of differential equation.Finally,abundant explicit expressions of exact traveling wave solutions are obtained by integrating along the special orbits in the phase portraits.The major works of this dissertation are as follows:In chapter one,we expound the development history,research status and the major research methods of solitons and partial differential equations,introduces basic steps of the method of bifurcations of dynamic systems and the major works of this dissertation.In chapter two,we study the exact traveling wave solutions of the(2+1)-dimensional coupled Burgers equation.By using the qualitative theory of differential equations,we give the bifurcation phase portraits of the integral constant g in different value ranges.We derive many new solitary wave solutions,kink wave solutions,periodic wave solutions and singular wave solutions through the special orbits in the phase portraits,which expand the previous research results.In chapter three,the bifurcation analysis of the Zakharov equation with a quantum correction is given,the phase portraits under different parameters and a series of new traveling wave solutions are obtained,including solitary wave solutions,periodic wave solutions and singular wave solutions.By taking the limit of partial periodic wave solutions,we show the process of periodic wave solutions evolving into solitary wave solutions and singular wave solutions.In chapter four,we utilize the method of bifurcations of the dynamic systems to study the exact traveling wave solutions of the modified Konopelchenko-Dubrovsky equation.We give the phase portraits and bifurcation analysis of the plane dynamic system,and then definite integral along the special orbits in the phase portraits,we derive a large number of exact traveling wave solutions when the exponent n is the general case,which generalize the solutions of this equation.The last chapter is the summary of the full dissertation and the prospect of future research.