Darboux Transformation And Conservation Laws Of A Class Of Semi-discrete Integrable System

Semi-discrete integrable systems are one of the important contents in nonlinear science,it has important applications in engineering,electricity,physics,life sciences and other fields,it can describe a lot of important natural phenomena,such as: particles in the lattice,vibration,electrical network,current,solid,discrete pulses in biological chains,and so on.In this paper,we construct a family of semi-discrete integrable systems by using Tu scheme.By constructing Darboux transformation,the exact solution of one of the semi-discrete integrable system in the hierarchy is obtained,and the infinite conservation laws of the system are established by using the Ricatti equation method.The full text is divided into three chapters,the first chapter is the introduction,the background and application of semi-discrete integrable systems and the exact solutions of several semi-discrete integrable systems are introduced.The second chapter is the preliminary knowledge,in which we introduce some basic concepts of semi-discrete integrable systems,review the Tu method of constructing semi-discrete integrable systems and present some contents about Darboux transformation and infinite conservation laws.The third chapter is the main results of this paper.We construct a family of semi-discrete integrable systems from the spectral problem,choose one of the system in the family,construct its Darboux transformation,generate the exact solution,and finally we obtain the infinite number of conservation laws of the system.