A Hierarchy of Coupled Korteweg–de Vries Equations and the Corresponding Finite-Dimensional Integrable System

Abstract

By introducing a 4×4 matrix spectral problem with three potentials, A hierarchy of nonlinear evolution equations are derived. An interesting 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, a new infinite-dimensional Hamiltonian system is presented, which is completely integrable in Liouville sense.