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

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

2009-11-03

arxiv.org/abs/0911.0607v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

54 pages, 2 figures

Scientific paper

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.

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