Hierarchies Of Nolinear Integrable Lattice Equations And Super-Integrable Systems

The major contents in this thesis include:hierarchies of integrable Lattice equations---discrete integrable systems or integrable nonlinear differential-difference equations---associated with 3-order spectral problems, integrable couplings of integrable Lattice systems and super-integrable systems and their super-Hamiltonian structures.Nonlinear integrable Lattice systems, 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 integrable Lattice systems have been obtained and discussed systematically. In Chapter Two, two 3-order isospectral problems are presented and two hierarchies of Lax integrable Lattice equations are derived from the previous spectral problems. Furthermore, it is shown that the two hierarchies are completely integrable in Liouville sense and possess bi-Hamiltonian structure respectively. In Chapter Three, two cases of integrable couplings have been discussed by means of the method of semi-direct sums of Lie algebras, one is an integrable Lattice system with three potentials associated with a 6-order discrete matrix spectral problem by coupling given an integrable Lattice system with one potential associated with a 2-order discrete matrix spectral problem, the other is an integrable Lattice system with six potentials associated with a 6-order discrete matrix spectral problem by coupling given an integrable Lattice system with three potentials associated with a 2-order discrete matrix spectral problem. 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 two continuous super-integrable systems. First, two continuous matrix spectral problems associated with two Lie superalgebras have been presented. Moreover, hierarchies of g-cKdV and mKdV equations have been derived, Finally, their super-Hamiltonian structures have been constructed respectively making use of the super-trace identity.