Darboux Transformation Of A Discrete Nonlinear Schr(?)dinger Equation And Its Exact Solutions

The paper is to study a discrete nonlinear Schro¨dinger equation(DNLSE).Webuild the Darboux Translation(DT) of the DNLSE and give its determinant expres-sion.With the help of the expression,the one-solition and2-solition solutions are ob-tained from zero seed. At the same time,another kind of solution is gained from non-zero seeds.There are four chapters in the paper.In chapter1,we describe the fundamen-tal theory of the discrete system and Darboux translation method.In chapter2,we firstget a coupled DNLSE from a Lax pair by the discrete zero curvature equation.With acertain reduction condition,the coupled equation can be transformed into a equationwhich is we need studying(A-L equation):Then a DT for the coupled DNLSE is derived with the help of the gauge transformationbetween the Lax pair.Third,the DT for the DNLSE is obtained through specific reduc-tion conditions.In chapter3,we discuss the application of the DT.We obtain the exactsolution of A-L equation from zero seed by DT.Moreover,with the aid of the Maple,thefigures of one-solition solution are given by the suitably chosen paramters.Then fromnon-zero seeds,we can also get the expression of new solutions.In the final chapter,wediscuss the N-fold Darboux translation and give its determinant expression.