Certain Study On Soliton Interaction In Fiber Communication And Other Fields

To research the real world,many interference factors should be considered,so that study of nonlinear systems is promoted.Studies on nonlinear systems in such fields as optical fibers,fluids and Bose-Einstein condensates will help researchers to understand the nonlinear phenomena.Theoretical research always concentrate on the controlled nonlinear evolution equations,then study the analytic solutions of those equations and predict the development of the solutions over time,which helps people to understand the nature and development law of things.The main contents of this article is arranged as follows:In chapter 1,we will introduce several types of nonlinear waves,i.e.,solitons,breathers and rogue waves,and the progress of these nonlinear waves.We also state certain methods involved in this paper,which are used to investigate the nonlinear evolution equations.Finally,we illustrate the arrangement of this paper.In chapter 2,we study a(2+1)-dimensional Konopelchenko-Dubrovsky system,and obtain soliton solutions with the determinant form for such a system via the Kadomtsev-Petviashvili(KP)hierarchy reduction method.By analyzing the property of these solu-tions and with different parameters,we obtain three kinds of solitons:bright solitons,anti-bright solitons and kink-type solitons.With analytic and graphic analyses,we present the elastic and inelastic interactions between the two solitons,as well obtain the conditions for the inelastic interaction.In chapter 3,we investigate a(2+1)-dimensional Davey-Stewartson system,which describes the evolution of surface water wave packets with finite depth.Using the KP hierarchy reduction and starting from the ? solutions with the determinant form,we obtain the periodic-wave solutions with some parameter conditions for such a system.Based on this periodic solutions,we present three kinds of breathers and a periodic wave with growing-decaying property.Taking the long-wave limit on the periodic-wave solutions,we derive the semi-rational solutions,and then lead to three types of lumps and a rogue wave with line profile.Thus,we draw the conclusion that:the rogue wave is the long-wave limit of the periodic wave,and the lump is the long-wave limit of the breather.In chapter 4,we study the coupled nonlinear Schrodinger equations with four-wave mixing term,which describe the optical solitons in a birefringent fiber.By applying a variable transformation,we map those equations into the standard coupled nonlin-ear Schrodinger equations(i.e.,the Manakov system).That is to say,solutions for the former equations can be seen as a linear superposition of solutions for the latter equation-s.Combining that variable transformation and the KP hierarchy reduction methods,we construct the bright-dark and dark-dark soliton solutions with determinant form.Combining that variable transformation and the Darboux-dressing transform method,we derive a semi-rational solutions which describe the interactions between the rogue waves and solitons,and between the rogue waves and breathers.Through analytic and graphic analyses,we draw that:(1)For the bright-dark solitons,we find that the soli-tons on the non-zero plane exhibit an oscillating feature,while the inelastic interaction between the bright-dark two solitons exhibits V-type and Y-type phenomena;(2)For the dark-dark solitons,we find that the soliton is of a periodic wave background,and through the asymptotic analysis,we obtain that the interaction between the dark-dark soliton is always elastic;(3)For the interactions between the rogue waves and solitons(or breathers),graphical analysis is performed.We get rich patterns of these nonlinear waves.In chapter 5,we take a(2+1)dimensional Date-Jimbo-Kashiwara-Miwa equation as an example,which is extended from the KP hierarchy.By using the Wronskian and pfaffian techniques,we obtain the Wronskian and Grammian solutions for such a equation,respectively,and also prove those two types of solutions(both of the N-th order determinant form).By choosing suitable parameters,we get the N-solitons with kink profile for both the Wronskian and Grammian solutions.In chapter 6,we investigate a three-component gross Pitaevskii(GP)equations,which are used to describe the spinor-1 Bose-Einstein condensate confined by an optical dipole trap with repulsive interaction.Through the binary Darboux transformation,we construct the vector dark-soliton solutions for those equations,and consequently obtain the kink type solitons and dark solitons.Compared with the existing dark solitons,we get a new W-type dark soliton.Among the interactions between the two solitons,we find that the interaction is always inelastic when a kink-type soliton is participate.In chapter 7,we will summarize the main contents of this paper,and propose the possible new research contents in the future.