Completion of the integrable coupling systems

Abstract

In this paper, we proposed an procedure to construct the completion of the integrable system by adding a perturbation to the generalized matrix problem, which can be used to continuous integrable couplings, discrete integrable couplings and super integrable couplings. As example, we construct the completion of the Kaup-Newell (KN) integrable coupling, the Wadati-Konno-Ichikawa (WKI) integrable couplingsis, vector Ablowitz-Kaup-Newell-Segur (vAKNS) integrable couplings, the Volterra integrable couplings, Dirac type integrable couplings and NLS-mKdV type integrable couplings.