The Study Of Lie Super Algebra And Nonlinear Evolution Equation Hierarchies

In this paper, super-integrable systems and their super-Hamlitonian structures, sym-metry constraints and binary nonlinearization for super-integrable systems, conservations laws and symmetry for Li equation hierarchy, the Geng equation with self-consistent sources are mainly discussed. The main contents in this paper are divided into four parts:1. From different types of basis of Lie super algebra.?(0,1) and their corresponding loop Lie super algebras, the super-isospectral problems of KN, GJ and Yang equation hierarchies are considered, then the super-KN, super-GJ, super-Yang equations hierarchies are obtained by using Tu scheme, lastly their super-Hamiltonian structures are given by using super trace identity. Furthermore, the first nontrivial equations of each equations hierarchy are given through computation.2. The binary nonlinearization method of integrable systems are generalized to super-integrable systems. Firstly the Bargmann symmetry constraint of the super NLS-MKdV equations hierarchy is considered, and the binary nonlinearization of the super NLS-MKdV equations hierarchy under this Bargmann symmetry constraint is given, then an implicit symmetry constraint of the super NLS-MKdV equations hierarchy is considered, and the binary nonlinearization of the super NLS-MKdV equations hierarchy under this implicit symmetry constraint is given; Secondly the Bargmann symmetry constraint of the super Guo equations hierarchy is considered, and the binary nonlinearization of the super Guo equations hierarchy under this Bargmann symmetry constraint is given, then an implicit symmetry constraint of the super Guo equations hierarchy is considered, and the binary nonlinearization of the super Guo equations hierarchy under this implicit symmetry con-straint is given; Finally the super classical-Boussinesq equations hierarchy is constructed, then a Bargmann symmetry constraint of the super classical-Boussinesq equations hierar-chy is considered, and the binary nonlinearization of this equations hierarchy under this symmetry constraint is given.3. The conservation laws of the Li equations hierarchy are given, then two symmetries of the Li equations hierarchy:?-symmetry and?-symmetry are given, and it is proved that the above two symmetries consists of a Lie algebra.4. The Geng equation with self-consistent sources is given by applying the source generation method which was proposed by Xingbiao Hu and Hongyan Wang to the Geng equation, then the Pfaffian Geng equation with self-consistent sources is also obtained by using the source generation method, and the commutative propertity between the source generation method and the Pfaffian method is verified for the Geng equation.