Applications of an anti-symmetry loop algebra and its expanding forms

Abstract

Constructing an anti-symmetry subalgebra A ̃ ̃ 2 of loop algebra A ̃ 2 gives the well-known Jaulent–Miodek (JM) hierarchy, the JM equation and its new Lax pair. Further, the Darboux transformation of the JM equation is deduced by anstaz method. By making use of a high-order loop algebra and Tu scheme, an expanding integrable model of the JM hierarchy is obtained. A direct expansion A ̄ 2 ∗ of loop algebra A ̃ ̃ 2 by considering the definition of Lie algebra is presented, which is used to establish two isospectral problems. It follows that corresponding two new integrable systems are engendered, which possess bi-Hamiltonian structures, respectively. Furthermore, a scalar transformation is applied to turn the loop algebra A ̄ 2 ∗ into its equivalent subalgebra A ̃ ̃ 1 of loop algebra A ̃ 1 . With the help of A ̃ ̃ 1 , another new high-order loop algebra G is constructed, which is used to obtain an expanding integrable model of one of two integrable systems presented.