Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti–Pempinelli–Tu hierarchy

Abstract

A kind of new Lie algebra is introduced whose corresponding loop algebra is followed to define. From which two isospectral problems are established whose compatibility condition gives rise to a coupling integrable coupling of the Boiti–Pempinelli–Tu (BPT) hierarchy. By making use of the quadratic-form identity, its bi-Hamiltonian structure is obtained. Finally, we reduce the coupling integrable coupling to obtain two integrable couplings of the BPT hierarchy. Furthermore, the well-known mKdV equation and a new coupled equation with two dependent variables are produced, respectively.