Characteristics Of Breather For Coupled Nonlinear Schr(?)dinger Equations

The coupled nonlinear Schr(?)dinger equation is the most important model in many nonlinear physical models,which can be describe many physical phenomena,such as the propagation of water waves in water tanks,dynamics of Bose-Einstein condensates with multiple components,the propagation of optical pulses with multiple degrees of freedom in optical fibers,and can even describe the rouge waves in the financial field.Studying coupled systems plays a very important role in many composite systems.In this thesis,we research the background of nonlinear systems and coupled nonlinear systems,and the development history of the soliton and its several types in the first chapter.Then,the process of solving coupled nonlinear Schr(?)dinger equation by means of the modified Darboux transformation were introduced.Finally,we study the exact analytical solution of the pair-transformation-coupled nonlinear Schr(?)dinger equation using the idea of Darboux transformation.As an example,two types of exact breather solution are given analytically by adjusting the parameters,i.e,Kuznetsov-Ma breather solution,Akhmediev breather solution.Firstly,the asymmetric and symmetric Kuznetsov-Ma breather solutions are studied.Three different types of Kuznetsov-Ma breather images are obtained with different values of parameters.And the images show their propagation characteristics in time-space coordinate system.Moreover,this thesis also clarify the formation mechanism of asymmetry and symmetry breather solutions.It is due to the particle number and energy exchange between the background and soliton which ultimately forms breather solutions.It can be described by the total nonuniform exchange.Then the exact asymmetric Akhmediev breather solution is studied.And the images also show their propagation characteristics in time-space coordinate system.Finally,we mainly study the rouge waves.Under the limit conditions,the density plots of the Akhmediev breather solution and Kuznetsov-Ma breather solution evolving into rouge waves are plotted,respectively.This is a local process from the periodic process of space and time to the background of plane wave.And finally from the breather solution to the rouge wave solution.In addition,the formation mechanism from Kuznetsov-Ma breather to rouge wave is also discussed.This mechanism arose from the transformation from the periodic total exchange into the temporal local property.