Investigations On Related Problems Of Integrable Differential-Difference Equations

For the sake of studying the formulation of discrete integrable systems and their many properties, the paper has listed some model of discrete integrable systems, and, some research such as the integrability, the Darboux transformation, infinite number of conservation laws and the nonlinearization have been investigated. 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 discrete 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 comparing with the continuous cases. In this paper, we formulated some discrete integrable systems and gave the corresponding lattice equations in 1+1 , 2+1 dimensions and associated Hamilton structure by means of the trace identity. To solve the obtained lattice equation, we construct the different Darboux transformation in light with the different spectral matrix by virtue of which the soliton solutions result, and the plots of these solutions are given resorting to the symbolic computation. As is well known that the property of possessing infinitely many conservation laws is very important for soliton equations. In this paper, we derived the conservation laws of 1+1, 2+1 dimensional differential-difference equations on the base of discrete spectral problems through a direct method. At last, the nonlinearization technique has been used for the discrete integrable systems to seperate an infinitely dimensional integrable system into two finite dimensional integrable systems.