The Study Of Properties And Solutions To Higher-Order Nonlinear Schr(o|¨)dinger Equations

With the qualitative theory of differential equations, this article studies a type of higher-order nonlinear Schrodinger equations in optical soliton field, gives a qualitative analysis of this equation, and derives the solutions via first integral method. This article obtains the solutions of a generalized Schrodinger-Boussinesq equation and a higher-order nonlinear Schrodinger equation through the generally projective Riccati equation method. This paper is arranged as follows:Chapter2focuses on a type of the high-order nonlinear Schrodinger equations in optical soliton field. With the traveling wave transformation and qualitative theory, ten different kinds of branches and the corresponding phase portrait diagrams are got. By observing the phase diagrams, there exists saddle points, sharp points and centers of the equations. Further analysis has been given of the branches of higher-order nonlinear Schrodinger equations when there exists five singularities of the equation. Combining the energy orbital distribution map, we find that corresponding to different values, of the equation there exists homoclinic branch, saddle-saddle point heteroclinic branch and the periodic orbit that the center corresponds to. As there exists first integral of the equation, via first integral method, we get the solitary wave solutions, kink wave solutions and periodic solutions of the equation corresponding to the orbit, shows the figures of the solutions itself and the solutions that parameters get the values around the critical point, and analyzes the reasons that the solutions changes around the critical point.In chapter3, through the generally protective Riccati equation method, the solutions of a generalized Schrodinger-Boussinesq equation and a higher-order nonlinear Schrodinger equation are obtained. Compared with the traditional Riccati method, with the generally projective Riccati equation method we can get hyperbolic function solutions, trigonometric function solutions, rational so-lutions and various forms of solutions. The applicable scope and the solutions have been greatly improved compared with the original method. With the help of Mathematica, new hyperbolic function solutions, trigonometric function solutions and rational solutions of the two equations are derived.