Several Integrable Lattice Equations And Properties

Main content of the paper consists of establishment of the Hamiltonian structure,conservation laws,Integrable coupling system of integrable lattice equations and Construction and Application of Darboux transformation.In the first chapter,production of soliton,development of the soliton theory and integrable system,application in science and technology as well as soliton's research meaning together with main content of the paper.In the second chapter,a discrete 2*2 spectral problem is introduced,a hierarchy of Lax lattice equations are obtained by the discrete zero curvature equation,the discrete Hamiltonian structure of the lattice equation is obtained by the trace-identity,as a result,the lattice equations are Liouville integrable.In the third chapter,a discrete 3*3 spectral problem is proposed,a hierarchy of Integrable lattice equations in Lax sense are got by the discrete zero curvature equation,and infinitely many conservation laws of the first lattice equation are obtained by the method of Lax pair of isospectral problem.In the fourth chapter,a discrete 4*4 isospectral problem is introduced,Integrable coupling system of integrable lattice equations is obtained by the method of half straight and Lie algebra;a discrete 2*2 non-isospectral problem is constructed,a hierarchy of 2+1 non-isospectral integrable lattice equations are derived by discrete zero curvature equation in non-isospectral form.The fifth chapter consists of two sections,the first section briefly introduces construction of the Darboux transformation,the second section mainly states application of the Darboux transformation.The sixth chapter is summary and forcasting.