Component-trace identity for Hamiltonian structure of the integrable couplings of the Giachetti–Johnson (GJ) hierarchy and coupling integrable couplings

The integrable couplings of the Giachetti–Johnson (GJ) hierarchy are obtained by the perturbation approach and its Hamiltonian structure is given for the first time by component-trace identities. Then, coupling integrable couplings of the GJ hierarchy are worked out.