Bi-integrable couplings and tri-integrable couplings of the modified Ablowitz-Kaup-Newell-Segur hierarchy with self-consistent sources

Abstract

We discuss the bi-integrable couplings and tri-integrable couplings of the modified Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy with self-consistent sources from the enlarged matrix spectral problem and the enlarged zero curvature equations. By using the variational identities over non-semisimple Lie algebras consisting of block matrices, we construct Hamiltonian formulations for integrable couplings, bi-integrable couplings, and tri-integrable couplings.