A Method for Obtaining Integrable Couplings

By making use of the vector product in R3, a commuting operation is introduced so that R3 becomes a Lie algebra. The resulting loop algebra R-tilde3 is presented, from which the well-known AKNS hierarchy is produced. Again via applying the superposition of the commuting operations of the Lie algebra, a commuting operation in R6 is constructed so that R6 becomes a Lie algebra. Thanks to the corresponding loop algebra R-tilde3 of the Lie algebra R3, the integrable coupling of the AKNS system is obtained. The method presented in this paper is rather simple and can be used to work out integrable coupling systems of the other known integrable hierarchies of soliton equations.