The conservation of the Hamiltonian structures in the deformations of the Whitham systems

Abstract

We consider the construction of the deformed Whitham system for the KdV-equation in the one-phase case and investigate the conservation of the Hamiltonian properties in this situation. It is shown then, that both the Gardner - Zakharov - Faddeev and the Magri brackets give the deformed Dubrovin - Novikov brackets (the brackets of Dubrovin - Zhang type) for the deformed Whitham system constructed by our procedure. The general approach used in the paper gives a scheme for the averaging of the Poisson structures in the general situation.

54 pages, 2 figures

References in corpus (3)

• On Hamiltonian perturbations of hyperbolic systems of conservation laws, II: universality of critical behaviour
• Whitham systems and deformations
• Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type