A Hierarchy Of Coupled Korteweg-de Vries Equations And The Corresponding Finite-dimensional Integrable System

By introducing a 4x4 matrix spectral problem with three potentials, we derive a new hierarchy of nonlinear evolution equations. A typical equation in the hierarchy is a coupled KdV equation. It is shown that the hierarchy possesses the generalized bi-Hamiltonian structures with the aid of the trace identity. Through the nonlinearization of eigenvalue problems, we get a new finite-dimensional Hamiltonian system, which is completely integrable in Liouville sense. In the end, we obtain the involutive solution of the coupled KdV equation.