Liouville Integrability Conservation Laws And Darboux Transformation Of 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, ,integrable couplings, infinite number of conservation laws and the Darboux transformation 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 associated Hamilton structure by means of the trace identity. In addition, integrable coupling is a new and significant direct in soliton theory. 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 some differential-difference equations on the base of discrete spectral problems through a direct method. At last, to solve the obtained lattice equation, we construct the Darboux transformation in light with the spectral matrix by virtue of which the soliton solutions result.