An R-matrix Theory Of Multi-component Integrable Lattice Systems And The Multi-component CTL-RTL Hierarchy

The thesis dedicates on constructing multi-component coupled lattice systems and establishing an r-matrix theory for a class of multi-component coupled lattice systems.It consists of two parts.The first part is to establish an r-matrix theory of multicomponent coupled lattice systems.A class of matrix-valued quasi-difference operators with permutation matrices is introduced.It is shown that,with the usual commutator Lie brackets,these quasi-difference operators constitute a Lie algebra.By constructing two kinds of Lie sub-algebraic decompositions of the Lie algebra and applying the general theory of r matrix,we obtain an r-matrix mapping.Then,the lattice hierarchy with infinite matrix potentials or multi-components are obtained.On this basis,by the finite truncations of the quasi-difference operators,a family of multi-component coupled lattice systems including multi-component Toda hierarchy are abtained.We show that the obtained lattice systems have natural first Poisson structures(Hamilton structures).Then,we take the multi-component Toda lattice as an illustrative example to show that its second Hamilton structure can be obtained by Dirac reduction.The second part is dedicated to constructing combined Toda lattice and relativistic Toda lattice(CTL-RTL)in multi-components.The multi-component analogies of the first two members in the CTL-RTL hierarchy and their zero-curvature representations,as well as bi-Hamiltonian structures,are constructed.