Study On Exact Solution And Integrability Of High Order Optical Soliton Equation

High-order optical soliton equation is a kind of important mathematical model in nonlinear science, which can be used to descibe the nonlinear phenomenon in the field of plasma physics, optical fibers telecommunication, superconduct, etc. The research on the exact solution and integrability of high-order optical soliton equa-tion contributes to understand the essential characteristics of the relevant dynamic mechanism and scientifically explains the corresponding physical phenomenon. In this paper, we focus on high-order optical soliton equation, and systematically investigate the exact traveling wave solution and Painleve integrability of some nonlinear Schrodinger equations with the aid of symbolic computation.Recently, with the development of symbolic computation, many scholars have proposed and developed some direct algebraic methods for constructing solution to nonlinear evolution equation. In Chapter 2, we at first summarize research progress of direct algebraic methods, and devide direct algebraic methods into two parts:special function expansion method and subsidiary equation expansion method. Moreover, we extend a class of subsidiary equation expansion method in order to obtain new exact traveling wave solution of nonlinear evolution equation with the aid of symbolic computation.In Chapter 3, we summarize some transformation techniques in which non-linear Schrodinger equation is reduced to real equation. Then, using the extended subsidiary equation method proposed in Chapter 2, we investigate a three-order nonlinear Schrodinger equation and a coupled three-order nonlinear Schrodinger equations, under some parameter conditions, we obtain some exact traveling wave solutions.There are close relationship between the Painleve integrability and other inte-grability of nonlinear evolution equation. In Chapter 4, using the Kruskal simpli-fied algorithm, we perform the Painleve test for a three-order variable coefficient nonlinear Schrodinger equation and a four-order nonlinear Schrodinger equation. The results indicate that the three-order variable coefficient nonlinear Schrodinger equation is Painleve integrable under two sets of parameter conditions, thus we can derive two three-order variable coefficient nonlinear Schrodinger equations of Painleve integrable. The four-order nonlinear Schrodinger equation can not pass the Painleve test due to the complex resonances. It deserves further study to how to obtain more integrable models starting from high-order nonlinear Schrodinger equation.