Studies On Four Super(-symmetric)integrable Systems

Four super(supersymmetric,SUSY)integrable system are studied in this thesis.The Lax representation and the linear spectral problem are the starting point of our studies,and then the integrable properties such as conservation laws,symmetries,recursion operators and Hamilton structures are carried out.Firstly,a four-component super CH-type equation is proposed and studied.In order to obtain a fermion extension of the generalized multi-component CH-type equation proposed by Xia et al.,by directly constructing the linear spectral problem,we obtain a four-component CHtype equation as the compatibility condition.Infinitely many conservation laws for the equation are constructed,and the recursion operator is worked out from the eigenvalue problem satisfied by the spectral gradients,furthermore,it is factorized to yield two compatible Hamiltonian operators,and the bi-Hamiltonian structure of the equation is established.Secondly,a multi-component super KdV equation proposed by Kupershmidt is studied.For the nontrivial scalar case,infinitely many conservation laws and the Backlund transformation transformation for the equation are constructed by means of the deformation.After projective coordinates introduced,the B?cklund transformation is linearized to give a Lax representation.The Lax representation is extended to the general case so that the integrability of the system is confirmed.The CAC property of the super fully discrete system induced by the Backlund transformation is verified and it provides us a discrete zero-curvature representation.Thirdly,one case of the SUSY NLS equation is studied,namely the SUSY NLS-B equation.We compute the prolongation algebra of the equation,and by embedding it into the Lie superalgebra A(1,1),the matrix representation and the linear spectral problem are given.After rewriting the equation and the linear spectral problem into superfields,the recursion operator is derived and from it we obtain a Hamiltonian operator which establishes a Hamiltonian structure with odd degree.Finally,the SUSY third-order Burgers equation is considered.We compute the prolongation algebra of the equation,and by embedding it into the Lie superalgebra A(1,0),the matrix representation is given,resulting in the linear spectral problem.Using this linear spectral problem,we calculate some higher-order symmetries of the equation...