Integrable Lattice Equation Hierarchy And Some Properties

This article mainly has seven parts.The first part mainly describes the research and development of soliton theory,the application of soliton and its significance.In the second part of this paper,a new discrete matrix spectral problem is introduced,and an integrable family of lattice soliton equations is derived by using the representation method of discrete zero curvature equation.Then the Hamiltonian structure is established by the equation family generated by the discrete trace identities,and the Liouville integrability of the corresponding lattice system is proved.Finally,the infinite conservation laws of the family of equations are solved.In the third part,a discrete third order spatial spectral problem is introduced.Then a family of integrable equations are obtained by using the discrete zero curvature equation.Finally,by applying the direct method of isospectral problem,infinite conservation laws of integrable coupled equations are obtained.In the fourth part,a 2+1-dimensional second-order non-isospectral problem is constructed,and a family of non-isospectral integrable lattice equations are solved by using the discrete zero curvature equation.The fifth part uses the Fourier transform of the Lax pair to obtain a dyadic transformation of the integrable lattice equation.In the sixth part of this paper,we propose the famous symmetric constraint,which is used to decompose every lattice equation in the system by using integrable symplectic mapping and fully integrable finite dimensional Hamiltonian system.The seventh part is the summary and Prospect of the article.