Duality relation among the Hamiltonian structures of a parametric coupled Korteweg-de Vries system

Abstract

AbstractWe obtain the full Hamiltonian structure for a parametric coupled KdV system. The coupled system arises from four different real basic lagrangians. The associated Hamiltonian functionals and the corresponding Poisson structures follow from the geometry of a constrained phase space by using the Dirac approach for constrained systems. The overall algebraic structure for the system is given in terms of two pencils of Poisson structures with associated Hamiltonians depending on the parameter of the Poisson pencils. The algebraic construction we present admits the most general space of observables related to the coupled system. We then construct two master lagrangians for the coupled system whose field equations are theε-parametric Gardner equations obtained from the coupled KdV system through a Gardner transformation. In the weak limitε→ 0 the lagrangians reduce to the ones of the coupled KdV system while, after a suitable redefinition of the fields, in the strong limitε→ ∞ we obtain the lagrangians of the coupled modified KdV system. The Hamiltonian structures of the coupled KdV system follow from the Hamiltonian structures of the master system by taking the two limitsε→ 0 andε→ ∞.