Analytical Study Of Nonlinear Schrodinger Equations With Variable Coefficients By KP Hierarchy Reduction

There are a lot of complex nonlinear phenomena in various field-s.Researchers mainly establish nonlinear models to study the nonlinear phenomena in various fields.The nonlinear Schrodinger equation is one of the typical models to describe the nonlinear phenomena and reveal the nonlinear laws.With the symbolic computation,in this dissertation,we analytically study some nonlinear Schrodinger equations with variable coefficients which can be uesd in the field of fluid mechanics,biophysics,optical fiber communications and so on.The soliton solutions of these equations are calculated and analyzed,and the influence of variable coef-ficients on the propagation and interaction of solitons is discussed.The primary contents of this dissertation are as follows:In chapter 1,the background of this paper is introduced,including nonlinear Schrodinger equation and soliton theory.The main research methods are shown which are used in this dissertation.And the structure of the article is briefly introduced.In chapter 2,the vector M-coupled Schrodinger equation is discussed.For arbitrary M,this equation governs the propagation of M-self-trapped mutually incoherent wave packets in Kerr-like photorefractive media.Vi-a KP hierarchy reduction,the detailed derivation and proof processfor solving the dark soliton solution of M-coupling equation are given.In chapter 3,the nonlinear Schrodinger equation with variable coef-ficient coupling is studied,which can be used to describe the simultane-ous propagation of the M-field components in an inhomogeneous optical fiber.The bilinear form of the coupled nonlinear Schrodinger equation is obtained by Hirota bilinear method,and N-bright-dark soliton solutions are constructed by KP hierarchy reduction.Through asymptotic analysis and graphic analysis,the propagation,and interaction of solitons under d-ifferent amplification/absorption effects and group velocity dispersion are discussed,especially the inelastic interaction of solitons.In chapter 4,we study a(3+1)-dimensional nonlinear Schrodinger equation with distributed coefficients for the spatiotemporal optical soli-tons or light bullets in a diffractive nonlinear Kerr medium with anoma-lous dispersion.Based on the similarity transformation of this equa-tion,the N-dark soliton solutions in terms of the Gramian are constructed via the KP hierarchy reduction,where N is a positive integer.In addition,via graphic analysis,the propagation and interaction of solitons under the effects from different diffraction/dispersion coefficient and gain coefficient are also discussed.In chapter 5,we summarize the main work of this dissertation and look forward to the future research direction.