Darboux Transformation And Soliton Solutions Of Two Discrete Nonlinear Evolution Equations

In this paper,we investigate two discrete nonlinear evolution equations including a dis-crete Hirota equation and a discrete coupled nonlinear Schrodinger(NLS)equation.By making use of the Darboux transformation(DT)method,we present the discrete soliton solutions of the two equations.At the same time,based on the discrete spectral problem of the discrete coupled NLS equation,an infinite number of conservation laws are constructed.The structure of the present paper is organized as follows:In chapter one,we introduce the history of development of the soliton theory at first.Then,we expound the DT method and the conservation laws.Finally,the overall works and arrangements of this paper are summarized.In chapter two,we consider a discrete Hirota equation.By virtue of the discrete spectral problem of the discrete Hirota equation,we derive the N-fold DT,and prove the results completely.Then,using qn = 0 as the seed,we have derived the new discrete soliton solutions.At the same time,the graphics of the solutions of the discrete Hirota equation are plotted via Mathematica.In chapter three,we consider a discrete coupled NLS equation.By the discrete spectral problem of the discrete coupled NLS equation,we derive the N-fold DT,and prove the results completely.Then,by virtue of the DT constructed,some new discrete soliton solutions have been generated on the vanishing and non-vanishing backgrounds.At the same time,figures of the solutions are plotted.Finally,the Riccati equation is got according to the Lax pair,and an infinite number of conservation laws of the discrete coupled NLS equation are deduced with the help of the compatibility of the Riccati equation.In chapter four,we summarize the conclusions of this paper and look forward to the future.