A kind of nonisospectral and isospectral integrable couplings and their Hamiltonian systems

Abstract

Abstract We first introduce a Lie algebra g ˜ which can be used to construct integrable couplings of some isospectral and nonisospectral problems. As two applications of the Lie algebra g ˜ , the MKdV spectral problem is enlarged to an isospectral problem and the AKNS spectral problem is expanded to a nonisopectral problem. Then, two integrable couplings are obtained by solving an isospectral and a nonisospectral zero-curvature equations. We find that the two hierarchies that we obtain have bi-Hamiltonian structure of combinatorial form. Additionally, some symmetries and conserved quantities of the resulting hierarchy are investigated.