N-fold Darboux Transformation For Several Kinds Of Nonlocal Coupled Discrete Nonlinear Schr(?)dinger Equation

Soliton theory is one of the mainstream of nonlinear science.It has brought revolutionary content and new vitality to the study of solving nonlinear partial differential equations and nonlinear science,making it one of the main means of studying nonlinear equations.There are many methods to construct exact solutions of nonlinear equations,but Darboux transformation plays an important role in the nonlinear methods.At present,the nonlinear Schr?dinger equation has made great achievement.However,the high order discrete coupled Ablowitz-Ladik equation and the discrete PT-symmetric nonlocal coupled nonlinear Schr?dinger equation have less work.In view of the above problems,the main contents of this paper are as follows:In Chapter 2,we mainly study nonlocal coupled nonlinear Schr?dinger equation with the self-induced PT symmetric.We proceed from a 3×3 matrix Lax pairs and use Darboux transformation method to obtain the relationships between the old and novel Lax pairs.And we mainly derive the 1-soliton,2-soliton and N-soliton formulas through complex computation.Then we get some novel exact solutions,including the bright soliton,breather wave,rogue wave soliton.Meanwhile,the elastic interactions between the two solitons are displayed,and their amplitudes keep unchanged after the interaction except for the phase shifts.In Chapter 3,we mainly study the discrete coupled Ablowitz-Ladik equations and the nonlocal coupled discrete nonlinear Schr?dinger equation.In the first part,we explore the Darboux transformation and exact solution of the discrete coupled Ablowitz-Ladik equation for 4×4 Lax pairs.In the second part,the discrete nonlocal Schr?dinger equation is skillfully extended to the discrete nonlocal coupled Schr?dinger equation,that is,2×2 Lax pair is extended to the 3×3 Lax pair.Firstly,the corresponding equation is obtained by using the discrete zero curvature equation.Secondly,the transformation matrix is constructed and the relationship between the new Lax pair and the old Lax pair is obtained by using the Darboux transformation.Finally,through complex calculation,the formulas of 1-soliton,2-soliton and N-soliton are obtained.By using the drawing software and choosing appropriate parameters,we can see the elastic collision of two soliton solutions,and obtain bright soliton solutions,breather solutions and rouge wave solutions.These conclusions can be fully applied to the nonlinear wave in some electro-optical systems.