Investigations On Related Problems Of The Integrable Coupling Systems And The Exact Solutions Of Nonlinear Differential-Difference Equations

The major contents in this paper include:the integrable systems of some nonlinear differential-difference equations associated with two-order spectral problems and their Hamiltonian structures; integrable couplings of nonlinear differential-difference equations and their Hamiltonian structures; the investigations of the exact solutions to some nonlinear differential-difference equations.Nonlinear differential-difference equations, shown to be an effective tool used for describing and explaining the nonlinear phenomena, have become the focus of common concern in recent years, and lots of nonlinear differential-difference equations have been obtained and discussed systematically. In Chapter Two, two 2-order isospectral problems are presented and two hierarchies of Lax integrable nonlinear differential-difference equations are derived from the previous spectral problems. Furthermore, it is shown that the two hierarchies are completely integrable in Liouville sense and possess Hamiltonian structures respectively. In Chapter Three, three cases of integrable couplings have been discussed by means of the method of semi-direct sums of Lie algebras, one is a 4-order discrete matrix spectral problem by expanding the 2-order discrete matrix spectral problem given in Chapter Two, another two 4-order discrete coupling systems are obtained from the other 2-order discrete matrix spectral problem in Chapter Two. Moreover, their Hamiltonian structures and Liouville integrability have been discussed respectively by means of the discrete variational identity. Chapter Four is devoted to the study of the exact solutions to nonlinear differential-difference equations. First, the(G'/G)-expansion method is introduced, and then, the exact solutions to nonlinear differential-difference equations have been derived via this method and the figures of the solutions are presented by Maple program.