Analytic Study On The Solitons And Rogue Waves In Some Fields Such As The Plasmas And Optical Fiber Communications

Nonlinear wave propagation in such fields as the plasmas and optical fibers can be described by the nonlinear partial differential equations, such as the nonlinear Schrodinger (NLS) equations. One type of those waves is the envelope soliton, which appears because of the balance between the dispersion and nonlinearity. In addition to the envelope solitons, the NLS equations admit other types of nonlinear waves, such as the soli-tons on finite background. It is generally recognized that solitons on finite background include Akhmediev breathers, Kuznetsov-Ma solitons and Peregrine solitons. The Peregrine soliton is a solution of the NLS equation with localization in both coordinates, and can be viewed as the mathematical description of a rogue wave. Here, with the analytic meth-ods and NLS-type equations, the solitons and rogue waves in some fields such as the plasmas and optical fiber communications will be studied.The results could be useful for understanding the generation mechanism and physical properties of solitons and rogue waves in such fields as the plasmas and optical fibers. The main research of this dissertation can be summarized as:(1) Study on the vector optical rogue waves and modulation insta-bility (MI) in an isotropic medium. Investigation is based on the coupled NLS equations with negative coherent coupling. Through the Darboux transformation (DT), some vector rogue-wave solutions of such equations are derived. It can be seen that a rogue wave can split up, giving birth to two rogue waves. Then, with the Darboux-Dressing transformation,vector rogue-wave solutions and conditions for the existence of vector rogue-wave solutions are obtained. With the analytic results, the prop-erties of vector rogue waves are discussed. In addition, the MI of such equations are studied. It is shown that if MI appears, it is of baseband type only.(2) Study on the triple Wronskian vector solitons and rogue waves of the coupled inhomogeneous NLS equations in the inhomogeneous plasma.With the non-isospectral Ablowitz - Kaup - Newell - Segur (AKNS) sys-tem and triple Wronskian identities, such equations are proved to admit the triple Wronskian vector soliton solutions. Effects of the parameters on the amplitude and velocity of the soliton are discussed. By virtue of the DT, the vector rogue-wave solutions of such equations are obtained.Influence of the parameters on the rogue waves is presented.(3) Based on the coupled third-order NLS equations, vector dark soli-tons and anti-dark solitons in the birefringent fibers are studied. With the Hirota method and symbolic computation, vector dark soliton solu-tions and anti-dark soliton solutions are derived. Parametric conditions for the existence of such vector solitons solutions are constructed, which are related to the third-order dispersion and self-steepening. Thorugh the graphical analysis, the propagation and collisions of vector dark solitons and anti-dark solitons are discussed.(4) Study on the breathers and rogue waves associated with the fifth-order dispersion in optical fibers. Investigation is based on the the fifth-order NLS equation. Through the DT, the Akhmediev breather,Kuznetsov-Ma soliton and rogue-wave solutions of such equation are ob-tained. Effects of the coefficients of the fourth-order dispersion, and of the fifth-order dispersion, on the properties of Akhmediev breather-s, Kuznetsov-Ma solitons and rogue waves are discussed.(5) Based on the nonautonomous Gross-Pitaevskii equation, the nonau-tonomous solitons and rogue waves in a Bose-Einstein condensate with an external potential are investigated. With the non-isospectral AKNS system and double Wronskian identities, such equations are proved to ad-mit the double Wronskian vector soliton solutions. Influence of the linear and harmonic potentials on the bound states between two matter-wave solitons is discussed. With the generalized DT, the first- and second-order rogue-wave solutions of that equation are derived. Effects of the linear and harmonic potentials on the background density, peak height and width of the rogue wave are discussed.(6) Study on the soliton excitations and collisions in the Alpha he-lical proteins. Investigation is based on the three-coupled fourth-order NLS equations and the three-coupled discrete NLS equations. For the three-coupled fourth-order NLS equations, with the Bell polynomials and Hirota method, bilinear forms and soliton solutions for such equations are obtained. Key point lies in the introduction of auxiliary functions in the Bell-polynomial expression. For the three-coupled discrete NLS equations, with the Bell polynomials and mixing variables, the N-soliton solutions and bilinear Backlund transformation of such equations are ob-tained. Based on the analytic results, propagation and interactions of the bound-state solitons are studied.