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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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Abstract

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
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.
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:40
Last Modified: 23 Sep 2013 18:32
URI: http://epubs.surrey.ac.uk/id/eprint/1388

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