Integrable Couplings, Variational Identities and Hamiltonian Formulations

Wen-Xiu Ma, Jinghan Meng, Huiqun Zhang

Abstract


We discuss Hamiltonian formulations for integrable couplings, particularly bi- and tri-integrable couplings,based on zero curvature equations. The basic tools are the variational identities over non-semisimple Lie algebrasconsisting of block matrices. Illustrative examples include dark equations and bi- and tri-integrable couplings of theKdV equation and the AKNS equations, generated from the enlarged matrix spectral problems. The associated variationalidentities yield bi-Hamiltonian formulations and hereditary recursion operators, thereby showing their Liouvilleintegrability.

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