Applications Of Lie (Super) Algebras To Integrable Systems And Their Hamiltonian Structures

There are three parts in this thesis.Part 1 devotes to some soliton integrable hierarchies and their Hamiltonian structures.First, from the Lie algebra B2 and another Lie algebra constructed by B2, we choose two spectral matrices satisfying Tu scheme, and construct two new soliton integrable hierarchies with Hamiltonian structures. Then, we consider two bases of the Lie algebra so (4), obtain two different soliton integrable hierarchies which can be reduced to the hierarchy of so(3),and find their relation. In addition, according to the isomorphism between the Lie algebras so(4) and su(2) (?) su(2), we get the relation between their corresponding hierarchies. Finally,from Levi theorem of Lie algebra, we construct bi-integrable couplings and tri-integrable couplings of the generalized AKNS hier?rchy with self-consistent sources.Part 2 studies some super soliton integrable hierarchies and their super-Hamiltonian structures. First,for the Lie superalgebra spl(2,1),we obtain a class of (1+1)-dimensional super soliton integrable hierarchies with Hamiltonian structures by the super-trace identity,and get three kinds of Darboux transformations of spectral matrices. Using TAH scheme, we generate a class of (2+1)-dimensional super soliton integrable hierarchies of the Lie superal-gebra spl(2,1) and their super-Hamiltonian structures. Then, we give super soliton integrable hierarchies and their super bi-Hamiltonian structures of the Lie superalgebras osp(2,2) and spo(2,2), respectively. And these super soliton integrable hierarchies can be reduced to the super AKNS hierarchy. According to spo(2,2) (?) osp(2,2) and spo(2,2) (?) sl(1, 2), we obtain the relation between their corresponding super soliton integrable hierarchies. Finally, we get the super AKNS hierarchy of B(0,n) and its conservation laws by the construction of the affine Lie superalgebra B(0, n)(1).Part 3 dedicates to a Lie algebra of zero curvature equations for bi-integrable cou-plings systems. First, we discuss a Lie algebra of continuous zero curvature equations for bi-integrable couplings systems, and apply the results to the generated isospectral and non-isospectral soliton hierarchies associated with the Lie algebra so(4). Then, we consider a Lie algebra of discrete zero curvature equations for bi-integrable couplings systems, and apply them to the generated isospectral and nonisospectral Toda hierarchies.