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Multi-symplectic structures and wave propagation

Bridges, Thomas J. (1997) Multi-symplectic structures and wave propagation the <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, 121 (01). pp. 147-190.


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A Hamiltonian structure is presented, which generalizes classical Hamiltonian structure, by assigning a distinct symplectic operator for each unbounded space direction and time, of a Hamiltonian evolution equation on one or more space dimensions. This generalization, called multi-symplectic structures, is shown to be natural for dispersive wave propagation problems. Application of the abstract properties of the multi-symplectic structures framework leads to a new variational principle for space-time periodic states reminiscent of the variational principle for invariant tori, a geometric reformulation of the concepts of action and action flux, a rigorous proof of the instability criterion predicted by the Whitham modulation equations, a new symplectic decomposition of the Noether theory, generalization of the concept of reversibility to space-time and a proof of Lighthill's geometric criterion for instability of periodic waves travelling in one space dimension. The nonlinear Schrödinger equation and the water-wave problem are characterized as Hamiltonian systems on a multi-symplectic structure for example. Further ramifications of the generalized symplectic structure of theoretical and practical interest are also discussed.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Bridges, Thomas
Date : 1 January 1997
Additional Information : Published in the Mathematical Proceedings of the Cambridge Philosophical Society, Volume 121, Issue 01, pp. 147-190. © 1997 Cambridge Philosophical Society. Reprinted with permission.Click here to visit the journal site.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:40
Last Modified : 16 Jan 2019 16:21

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