The Integrability And Analytical Study Of Several Classes Of Self-dual Network Equations

In recent years,the research of many scholars has gradually changed from continuous integrable system to discrete integrable system.Nonlinear differential-difference equations,taken as the spatially discrete analogues of nonlinear partial differential equations,have been used to describe certain physical phenomena.For example,nonlinear self-dual network equation may describe the propagation of electrical signals in a nonlinear,lumped,self-dual ladder-type electric circuit,so it is of great significance to study integrability properties and analytical solutions related to this equation due to its special physical background.In this paper,the discrete zero-curvature equation and discrete Darboux transformation are used to investigate the integrability properties and exact analytical solutions and their corresponding dynamical behaviors of several types of discrete integrable systems related to nonlinear self-dual network equation on the basis of Lax integrability in the sense of 2 × 2 matrix linear spectrum problem.The main contents include the following two aspects:(1)The integrability,analytical solutions and related modulation instability of four kinds of nonlinear self dual network equations are studied.Firstly,based on the spatial part of the known linear spectral problem,the integrability properties such as the integrable hierarchy,Lax pair and infinite conservation law are studied by using the discrete zero-curvature equation.Then,based on the new obtained Lax pair,the discrete N-fold Darboux transformations and generalized(m,N-m)-fold Darboux transformations of four kinds of equations are constructed,different types of soliton structures such as soliton,rational soliton,semi-rational soliton and mixed interaction structures are obtained.Through the asymptotic analysis of the exact solutions,the asymptotic state expressions are discussed before and after soliton collision,and the mathematical characteristics of rational soliton solutions are also analyzed.With the help of the computer symbol software Maple,the soliton structures and propagation phenomena are analyzed.The dynamic evolution and propagation stability of soliton solutions are discussed by numerical simulation with MATLAB;(2)The integrability,analytical solutions and dynamical behavior of Toda type lattice equation,i.e.modified exponential Toda lattice equation,related to nonlinear self-dual network equation are studied.Firstly,based on the known spatial part of linear spectral problem,the integrable hierarchy,Hamiltonian structure and Liouville integrability of this equation are given by using Tu scheme method.Then,based on the new obtained Lax pairs,the discrete generalized(m,N-m)-fold Darboux transformation is constructed to give different types of analytical solutions such as soliton solutions,rational solutions and semi-rational solutions and mixed solutions.The asymptotic state expressions of these solutions are studied by asymptotic analysis,and the mathematical characteristics of rational solutions are discussed.With the help of computer software maple and MATLAB,the elastic interaction phenomenon and propagation stability of soliton solutions are discussed by graphical analysis and numerical simulations.In particular,we find the new elastic interaction phenomena of kink solitons on the inclined plane background of this discrete equation.