The Soliton Solutions Of Several Discrete And High-dimensional Nonlocal Nonlinear Models

In this thesis,we mainly investigate several discrete and high-dimensional nonlocal nonlinear models with strong physical significance,including generalized discrete nonlinear Schr(?)dinger(NLS)model with variable coefficients,discrete coupled nonlinear Schr(?)dinger(NLS)model,discrete coupled nonlinear modified Korteweg-de Vries(mKdV)model and nonlocal nonlinear(2+1)-dimensional Schr(?)dinger-Maxwell-Bloch(SMB)model.By using the Darboux transformation(DT)algorithm and analyzing the infinite conservation laws,the integrability,the explicit solutions and the interactions of solitons are investigated.The structure of the present thesis is organized as follows:In chapter one,we first review the development history of nonlinear science and soliton theory,then expound the -symmetric operator theory,next,give the main ideas of DT algorithm and infinite conservation laws in soliton theory,and finally,the overall works of this thesis are summarized.In chapter two,the generalized discrete NLS model with variable coefficients is studied.The new discrete multi-soliton solutions with zero initial solutions are calculated by building variable coefficients discrete N-fold DT algorithm.By adjusting and controlling parameters,the graphical simulations of soliton solutions are carried out by using Mathematica,and the transmission dynamics and elastic collisions of the discrete solitons when N=1 and N=2 are discussed.In chapter three,the discrete coupled NLS model is studied.Based on discrete N-fold DT algorithm,we derive new discrete multi-soliton solutions for this model and verify that the interactions of solitons are elastic through the asymptotic analysis method with symbolic computation.The Riccati equation is obtained from the Lax pair of the model,then discrete form of infinite many conservation laws is derived by means of compatibility condition.In chapter four,the discrete coupled nonlinear mKdV model is studied.Through symbolic computation the discrete spectral problems are analyzed,the discrete N-fold DT algorithm is constructed,discrete one-soliton and two-soliton solutions in the form of Vandermonde-like determinants are derived and some dynamic behaviors of those solitons discussed graphically.Finally,infinite many conservation laws of the discrete coupled nonlinear mKdV model are deduced with the help of the compatibility of Riccati equation.In chapter five,the nonlocal nonlinear(2+1)-dimensional SMB model is studied.First of all,the analytic algorithm of the one-fold DT is derived from the spectral problem satisfied by the model,and the complete proof process is given.Secondly,the periodic waves solutions,soliton solutions and complexiton solutions of the model under different initial non-zero solutions are obtained.By adjusting and controlling parameters,various types of analytic solutions are simulated graphically by using Mathematica,and the transmission dynamics and elastic collisions of solitons are discussed.In chapter six,we summarize the conclusions of this thesis and look forward to the future.