Soliton Solutions And Some Related Properties Of Nonlinear Evolution Equations Based On Symbolic Computation

With the development of mathematics, physics and other subjects, people have deep understanding and application for the linear system, but the linear system is only for the approximate linear abstract of the complex phenomena. In the nature science and engineering technology, most phenomena and models can not be described only by the linear system, which encourages people to pay much attention to study the nature of the phenomena and makes the non-linear science emerge and develop fast. As nonlinear factors can be found in most fields of natural science and social science, and nonlinear system can il-lustrate different complex phenomena more accurately than linear system, so the nonlinear science plays a more and more important role in the research and application of theory and develops rapidly in various fields, such as fluid mech-anism, optical fiber communication, plasma physics and so on. Soliton, as a branch of the nonlinear science, can describe the propagation, collision and en-ergy exchange of nonlinear waves. Recently, soliton theory have developed fast and got a lot of attention. The primary basis is to obtain the soliton solutions, and now there are many methods to derive the soliton solutions, such as Hirota bilinear method, Backlund transformation, Bell-polynomial technique and Dar-boux transformation. Using the asymptotic analysis and graphical simulation, and combining some of the above methods, we investigate the soliton solutions and the properties of the solutions of some nonlinear evolution equations ana-lytically. The main work of this dissertation are as follows:(1)An extended modified Kadomtsev-Petviashvili equation, which can describe the propagation of the ion-acoustic waves in a plasma with non-isothermal electrons, is investigated analytically. Based on the properties of the Bell polynomials, bilinear form and Backlund transformations are obtained. With the Hirota method, we derive the soliton solutions of the equation. Soli-ton propagation and interaction are discussed graphically:the parametric con-ditions, through which the bell-shaped solitons, the anti-bell-shaped solitons or shock waves can emerge, are given; the influence of coefficient parameters on the soliton propagation and interaction is discussed; different types of elastic and inelastic interactions are presented.(2)Under investigation is a (2+1)-dimensional nonlinear evolution equation generated via the Jaulent-Miodek hierarchy. Through the Bell polynomials, the Hirota method and symbolic computation, bilinear forms and Backlund trans-formations are derived and N-soliton solutions are constructed. We discuss soliton propagation and interaction analytically and graphically and show the condition that cause the parallel elastic interaction.(3)For a (3+1)-dimensional breaking soliton equation which describes the reacting mixtures and shallow water waves, we obtain one-, two-, three-and N-shock-wave solutions of the equation. Based on those solutions, we discuss the propagation and collision of the shock waves analytically and graphically on different planes and show the oblique and parallel elastic collisions.(4)Considering the propagation of orthogonally-polarized optical waves in an isotropic fiber, we analyze the coherently-coupled nonlinear Schrodinger equations. Via the Hirota method and symbolic computation, bilinear forms are obtained and bright one-and two-soliton solutions are derived. Through the asymptotic analysis, soliton propagation and interaction are investigated. With the graphical simulation, head-on and overtaking elastic interactions are displayed, and the influence of the coefficient on soliton amplitude is discussed.(5)The (3+1)-dimensional coupled nonlinear Schrodinger equations for an optical fiber with birefringence are investigated. With the Hirota method and symbolic computation, bilinear forms of the equations are derived via an aux-iliary function, and the bright one-and two-soliton solutions are constructed. Soliton propagation and interaction are investigated analytically and graphical-ly:the single-peak and two-peak solitons are presented; different properties of the elastic and inelastic interactions are analyzed.