(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang-Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.