Research On The Pole Solutions Of The Discrete Nonlinear Schrodinger Equations

With the development of nonlinear science,a large number of nonlinear evolution equations have arisen and they play an important role in different physical backgrounds.The soliton solutions of nonlinear Schrodinger equation have been widely studied,but the multi-pole solutions have attracted little attention.This dissertation is devoted to the study of the dynamic properties of the discrete nonlinear Schrodinger(DNLS)equation and its solutions.In the first part,we use the bilinear method to construct the soliton solutions of the single-component DNLS equation;Then,the multi-pole solutions are derived from its multi-soliton solutions via some limit technique;Also,the asymptotic analysis is performed for such multi-pole solutions(MPSs)by considering the balance between exponential and algebraic terms,and the dynamic properties of the asymptotic solitons of the pole solitons are analyzed.In the second part,we study the discrete coupled DNLS equation,we construct the bright-bright and bright-dark two-soliton solutions for the focusing-focusing and focusing-defocusing cases by the bilinear method.Similarly,on the basis of the soliton solutions,the corresponding MPSs are also obtained by using the limit technique and the asymptotic analysis is carried out to study the dynamic properties of the asymptotic soliton.Based on the above research,we find that the asymptotic solitons of the MPSs of DNLS equation have the amplitudes and the interactions are elastic.But in contrast to the common multi-soliton solutions,the MPSs have two types of asymptotic solitons:one lies in a straight line of the space-time plane and the propagation velocity does not change with time;the other one lies on a curve in the space-time plane and the propagation velocity changes with time.In addition,the relative distance between asymptotic solitons increases logarithmically with the time,but the separation acceleration decreases exponentially with the increase of relative distance.