Discrete Integrable System And Its Darboux Transformation,Nonlinearization,Conservation Laws

For the sake of studying the formulation of discrete integrable systems and their many properties, we listed some model of discrete integrable systems and some research has been done such as their integrability, the solutions of corresponding soliton equations, infinitely many conservation laws and the nonlinearization in this paper. In fact the discrete integrable system can be used as the model of some problem in physics, chemistry and biology, for example the Toda lattice equation and the Volterra lattice equation. So it is important to study the dixcrete integrabe systems. However it is difficult to find new integrable system which is different from that of continuous integrable systems and there is fewer papers compared with the other case. In this paper we formulated some discrete integrable systems and gave the corresponding lattice equations and its Hamilton structure by means of the trace identity. To solve the lattice equation, we gave the different Darboux transformation matrix get some new exact solutions. We have known that some soliton equations have infinitely many conservation laws and many ways have been given to get it. In this paper we derived the conservation laws on the base of discrete spectral problem. At last the nonlinearization technique has been used in the discrete integrable systems to transform a infinitely demission integrable system into two finite demission integrable systems..