Decoupling Of Soliton Solutions Of Non-local Schr(?)dinger Equations By Darboux Transform And Bilinear Method

At present,the application of Darboux transform to solve soliton equation is still a hot research problem.Because the Darboux transform is more complex,most scholars only study the transformation of 2×2 Lax pairs.In recent years,the research results of non-local coupled Schr?dinger equation are less.Bilinear method is used to solve local soliton equations,which can simplify part of the calculation process and make it easier to find soliton solutions.However,a bilinear method is rarely used to solve the soliton solution of coupled non-local Schr?dinger equation.In view of these problems,this paper studies them from the following aspects.In the second chapter,the Darboux transformation of the coupled non-local nonlinear Schr?dinger equation with 3×3 spectrum problem is studied.Starting from a special Lax pairs,the CNNLS equation is constructed.Then,we obtain the one-and two-and N-soliton solution formulas of the CNNLS equation with N-fold Darboux transformation.Based on the obtained solutions,the propagation and interaction structures of these multi-solitons are shown,the evolution structures of the one-dark and one-bright solitons are exhibited with N=1,the overtaking elastic interactions among the two-dark and two-bright solitons are considered with N=2.The obtained results are different from the solutions of the local nonlinear equations.Some different propagation phenomena can also be produced through manipulating multi-soliton waves.In the third chapter,we investigate the Darboux transformation of mixed local-nonlocal vector Schr?dinger equations with a 3×3 spectral problem.Starting from a special Lax pairs,the mixed local-nonlocal vector Schr?dinger equations are constructed.We obtain the one-and two-and N-soliton solution formulas of the mixed local-nonlocal vector Schr?dinger equations with N-fold Darboux transformation.Based on the obtained solutions,the propagation and interaction structures of these multi-solitons are shown,the evolution structures of the one-solitons are exhibited,the overtaking elastic interactions among the two-breather solitons are considered.We find that unlike the local and nonlocal cases,the mixed local-nonlocal vector Schr?dinger equations have some novel results.In the fourth chapter,bilinear method is used to solve coupled non-local nonlinear Schr?dinger equations.Firstly,from the nonlinear equation,the transformation of the correlation variable is applied,and the original nonlinear equation is transformed into the double linear equations by introducing the bilinear operator.Secondly,the partial differential equations are constructed and solved.Finally,the 1-soliton solution and2-soliton solution of coupled non-local nonlinear Schr?dinger equation are obtained.And the Maple software is applied to draw the multi-soliton solution.The results in this paper might be helpful for understanding some physical phenomena in plasmas.