On the Hamiltonian structure of ion-acoustic plasma waves and water waves in channels

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Abstract
It is shown that the Hamiltonian structure of ion‐acoustic waves and channel waves may be used to derive the Hamiltonian structure of the Korteweg–de Vries equation and its higher‐order corrections. The Hamiltonian approach used here is more systematic and less laborious than standard methods for deriving the Korteweg–de Vries equation. It is also more revealing. In particular, it is shown that the Poisson bracket of the corrected equations equals the Korteweg–de Vries Poisson bracket at every order. It is also shown that the corrected equations become nonlocal at sufficiently high order.