Three kinds of coupling integrable couplings of the Korteweg–de Vries hierarchy of evolution equations

Abstract

We introduce three kinds of column-vector Lie algebras Ls(s=1,2,3). By making invertible linear transformations we get the corresponding three induced Lie algebras. According to the defined loop algebras L̃s of the Lie algebras Ls(s=1,2,3), we establish three various isospectral problems. Then by applying Tu scheme, we obtain three different coupling integrable couplings of the Korteweg–de Vries (KdV) hierarchy and further reduce them to three kinds of explicit coupling integrable couplings of the KdV equation. One of the coupling integrable couplings of the KdV hierarchy of evolution equations possesses Hamiltonian structure obtained by using the quadratic-form identity and it is Liouville integrable.