Some Algebraic Relations Common to a Set of Integrable Partial and Ordinary Differential Equations.

摘要

In recent years, a vareity of completely integrable Hamiltonian systems, both in ordinary and partial differential equations, have been subject to intensive scrutiny. Examples are the Toda and Kac-Moerbeke systems in ordinary differential equations, and their respective continuum limits, the Boussinesq and Korteweg-de Vries equations in partial differential equations. Of interest, for at least a formal understanding of the above mathematical phenomena, is the existence of relations common to all these phenomena, for instance, the so called Backlund transformations, or the associated Lax isospectral formulation of the above equations. This paper exhibits a common method for the construction of Lenard relations, which yield along with Gelfand-Dikii type operator trace formulas, an explicit recursive construction of the hierarchy of integrable systems associated with each of the above systems. In addition, an operator-valued function is constructed which yields the infinitesmal generators of the Lax type isospectral deformations associated with the above mentioned hierarchies of systems.