Integrable Properties And New Soliton Solutions Of Discrete Nonlocal NLS Equation

Nonlocal nonlinear equations have arisen in a variety of fields,such as fluid mechanics,optics,condensed matter physics,and high-energy physics,and have high research value and a wide range of applications.This dissertation is devoted to studying infinite conservation laws,N-th Darboux transformation and new non-singular soliton solutions of the discrete nonlocal nonlinear Schr?dinger(NNLS)equation.First,combining the spatial discrete and time development parts of the Lax pair,we construct two families of infinite conservation laws of discrete NNLS equations based on the negative and positive power series expansion in the spectral parameter,respectively.Meanwhile,we give the explicit forms of the first three conservation laws.Second,we provide a rigorous proof for the N-th Darboux transformation,that is,the transformed eigenfunction and potentials completely satisfy the Lax pair.In particular,by using the Lax pair and determinant properties,we develop new identity relations to prove that the constraint relations among potentials hold after the transformation.Additionally,we obtain the determinant representation of N-th iterated solution by the iterative algorithm of Darboux transformation.Third,taking the hyperbolic tangent function solution as the seed,we construct the first-order exponential soliton solution of the self-focusing discrete NNLS equation by using the Darboux transformation.Moreover,we discuss the dynamical behavior of the solution through graphics,revealing that the solution admits the soliton interaction properties similar to those in the the self-defocusing discrete NNLS equation.