With Discrete Variables Of Integrable Models And B (?) Backlund Transform

In this thesis, our objective is mainly focused on studying the decomposition of the discrete models with two variables and the inner relations between continuous soliton equation and discrete one.Four discrete eigenvalue problems are discussed in detail. Starting from the fundamental communicative identity, the integrable symplectic map with its conserved integrals is obtained through the nonlinearization procedure of eigenvalue problems; then with the help of compatibility, we have arrived at two double-discrete models relating to continuous AKNS model and KN model; based on these results, we make the flow straightened out, and give out the explicit solutions for the flow of two double-discrete models in Abel-Jaccobi coordinates.At the same time, starting from continuous AKNS model and KN model, we have constructed their Backhand transformations respectively with the help of compatibility, which are properly generated by the discrete soliton equations associated with four discrete eigenvalue problems.In the end of this thesis, we make out the inner relation between discrete Ablowitz-Ladik model and a continuous model, the former generates the backlund transformations of the latter.