A Class of Coupled KdV Systems and Their Bi-Hamiltonian Formulation

Bi-Hamiltonian formulation is significant for investigating integrable properties of nonlinear systems of differential equations [1] [2] [3]. Many mathematical and physical systems have been found to possess such kind of bi-Hamiltonian formulation. There are two important problems related to bi-Hamiltonian theory. The one is which kind of systems can possess bi-Hamiltonian formulation and the other one is how we construct bi-Hamiltonian formulation for a given system if it exists. There has been no complete answer to these two problems so far, although a lot of general analysis for bi-Hamiltonian formulation itself has been made. However we can make as many observations on structures of various bi-Hamiltonian systems as possible, through which we may eventually find a possible way to the final end. With such an idea or a motivation to enhance our understanding of bi-Hamiltonian formulation, we would like to search for new examples of bi-Hamiltonian systems among coupled KdV systems and their higher order partners. There are already some theories which allow us to do that. For instance, we can generate soliton hierarchies by using decomposable hereditary operators [4] or by using perturbation around solutions [5]. In this paper, we would just like to present some new concrete examples to satisfy the Magri scheme [1] by considering decomposable hereditary operators. Let us choose two specific matrix differential operators: