Integrable couplings, bi‐integrable couplings and their Hamiltonian structures of the Giachetti–Johnson soliton hierarchy

On the basis of zero curvature equations from semi-direct sums of Lie algebras, we construct integrable couplings of the Giachetti-Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi-integrable couplings of the Giachetti-Johnson hierarchy and their Hamiltonian structures by applying a class of non-semisimple matrix loop algebras consisting of triangular block matrices. Copyright (c) 2014 John Wiley & Sons, Ltd.