Construction Of The Generalized Hierarchy Associated With The Lie Super Algebra Osp(2,2)

In the field of soliton and integrable system,it is one of the hot topics to construct new integrable hierarchies.In the past few decades,based on a Lie algebra,the spectral problem was constructed.The compatible condition of the spectral problems yielded to the famous zero curvature equation.Thus,some integrable hierarchies would be constructed successfully.In this paper,starting from the Lie superalgebra osp(2,2),we will construct super AKNS and super Dirac hierarchies associated with the Lie superalgebra osp(2,2).The detailed studies are listed as follows:In the first chapter,we will briefly recall the development of soliton and integrable system,the sense of nonisospectral hierarchies,the significance of Lie superalgebra osp(2,2)and the main work of this paper.In the second chapter,based on the Lie superalgebra osp(2,2),we will choose a set of basis of the Lie superalgebra osp(2,2).A linear combination of the chosen basis matrices is regarded as the spatial part of the super AKNS spectral problem.By making use of the compatible condition of spatial and temporal parts,we derive the famous zero curvature equation.When the spectral parameter is independent of the time variable,we can obtain the isospectral super AKNS hierarchy.When the spectral parameter is dependent of the time variable,we can obtain the nonisospectral super AKNS hierarchy.Furthermore,we can derive the generalized nonisospectral super AKNS hierarchy associated with the Lie superalgebra osp(2,2).A similar research will be appeared in chapter 3.Another linear combination of the basis matrices is took as the spatial part of the super Dirac spectral problem.By the similar way,we derive the isospectral super Dirac hierarchy,the nonisospectral super Dirac hierarchy and the generalized nonisospectral super Dirac hierarchy associated with the Lie superalgebra osp(2,2).In the fourth chapter,based on Lie superalgebra osp(2,2),a generalized super AKNS spectral problem is considered.By using the compatible condition of the spectral problems,the zero curvature equation is obtained.Furthermore,we derive the generalized super AKNS hierarchy associated with the Lie superalgebra osp(2,2).By making use of the supertrace identity,the generalized super AKNS hierarchy can be rewritten as the super bi-Hamiltonian structure.A similar research will be appeared in chapter 5.We consider a generalized super Dirac spectral problem.By the similar way,we derive the generalized super Dirac hierarchy associated with the Lie superalgebra osp(2,2)and rewrite it as the super biHamiltonian structure.Some conclusions and discussions are listed in the last chapter.