Theory of integrable lattices

This thesis deals with the theory of integrable lattices in "solitons"throughout. Chapter 1 is a general introduction, which includesan historical survey and a short surrunary of the "solitons" theoryand the present work.In Chapter 2, we discuss the equivalence between two kinds oflattice AKNS spectral problems - one includes two potentials,while the other includes four. The two nonlinear lattice systemsassociated with those two spectral problems, respectively isalso proved to be equivalent to each other.In Chapter 3, we derive a class of nonlinear differential-differenceequations (NDDEs) and put them into the Hamiltonian systems.Their complete integrability are proved in terms of so called"r-matrix". In the end of this Chapter, we study the symmetryproperties and the related topics for lattice systems. In particular,we give detail for the Toda lattice systems.Chapter 4 is concerned with the Backlund transformations (BTs)and nonlinear superposition formulae (NSFs) for a class of NDDEs.A new method is presented to derive the generalized BTs andto prove that these BTs are precisely and really the auto-BTs.The three kinds of NSFs are derived by analysis of so called"elementary BTs".In Chapter 5, we investigate some relations between our latticesand the well-studied continuous systems. The continuum limitsof our lattice systems and the discretizations of the continuoussystems are discussed. The other study is about how we canconsider a BT of continuous systems as a NDDE and then how aBT of such a NDDE can be reduced to the three kinds of NSFsof the continuous systems.The last Chapter is a study of integrable lattices under periodicboundary conditions. It provides a mathematical foundationfor the study of integrable models in statistical mechanics.We are particularly interested in the lattice sine-Gordon andsinh-Gordon models. We not only prove the integrability ofthese models but also derive all kinds of classical phase shiftsand some other physically interesting relations.