Study On Traveling Wave Solutions For Several Kinds Of Lotka-Volterra Diffusive Equations And Nonlinear Schr(?)dinger Equations

In this thesis,we focus on the study of dynamical behavior and traveling wave solutions for several kinds of diffusive Lotka-Volterra equations and nonlinear Schr (?)dinger equations through the qualitative theory of differential equations,Gr (?)bner basis elimination method,resultant elimination method,computer algebra and symbolic calculation,phase portraits analysis of dynamic systems and the complete discrimination system method.The main contributions of the thesis are summarized as follows.In Chapter 3,we apply the qualitative theory of dynamical systems,the necessary and sufficient conditions for the existence of traveling wavefronts for the Fisher equation with a wave velocity of (8 are obtained via the definition and properties of traveling wave fronts(monotone traveling wave solutions).The obtained conditions improve the corresponding results in literature [88].And then,a class of two species Lotka-Volterra competitive diffusive Lotka-Volterra equations are considered,the necessary and sufficient conditions for the existence of traveling wave fronts connect the origin point with the positive equilibrium point as well as the positive equilibrium point with the boundary equilibrium point are obtained,which improve the corresponding results in references [91,93].Finally,we point out the printing errors in the paper[93],then we correct them.In Chapter 4,the two species Lotka-Volterra diffusive equations and the completely symmetric three species Lotka-Volterra diffusive equations are considered.With the help of Gr (?)bner bases elimination method,resultant elimination method and symbolic calculation,we construct the traveling wave solutions which connect the origin point with the boundary equilibrium point,the origin point with the positive equilibrium point and the positive equilibrium point with the boundary equilibrium point.A set of numerical examples for the open problem in reference [120] are given at the same time.Finally,the traveling wave solutions which connect the origin point with the positive equilibrium point and the positive equilibrium for the three species Lotka-Volterra diffusive equations are obtained.The obtained conditions improve the work of Mimura et al in the literature [115,116].In Chapter 5,we focus on the fractional coupled nonlinear Schr (?)dinger equation,by using the complete discrimination system method and symbolic calculation,we obtain the classification of all single wave solutions of this equation,which include trigonometric function solutions,hyperbolic function solutions,solitary wave solutions,rational function solutions and Jacobi elliptic function solutions.In order to further explain the propagation of the fractional coupled nonlinear Schr (?)dinger equation in nonlinear optics,two-dimensional and three-dimensional graphs are drawn.In Chapter 6,by some suitable traveling wave transformations,the Schr (?)dinger-Hirota equation is reduced to an autonomous planar dynamical system.By using the bifurcation theory of dynamical system and the method of phase plane analysis,combining with the Hamilton conserved quantity,integrating along different orbits via the Jacobi elliptic function method,a series of new traveling wave solutions for the nonlinear Schr (?)dinger-Hirota equation are obtained,which include periodic wave solutions,bell solitary wave solutions and kink wave solutions.Then by using the complete discrimination system method and symbolic calculation,the classification of single wave solutions for the nonlinear Schr (?)dinger-Hirota equation is given.Finally,the three-dimensional and two-dimensional graphs of the nonlinear Schr (?)dinger-Hirota equation are drawn,which further understand the propagation of the nonlinear Schr (?)dinger-Hirota equation in nonlinear optics.