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Dissipation in Hamiltonian systems: decaying cnoidal waves

Derks, Gianne and van Groesen, E. (1996) Dissipation in Hamiltonian systems: decaying cnoidal waves SIAM Journal on Mathematical Analysis, 27. pp. 1424-1447.


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The uniformly damped Korteweg--de Vries (KdV) equation with periodic boundary conditions can be viewed as a Hamiltonian system with dissipation added. The KdV equation is the Hamiltonian part and it has a two-dimensional family of relative equilibria. These relative equilibria are space-periodic soliton-like waves, known as cnoidal waves. Solutions of the dissipative system, starting near a cnoidal wave, are approximated with a long curve on the family of cnoidal waves. This approximation curve consists of a quasi-static succession of cnoidal waves. The approximation process is sharp in the sense that as a solution tends to zero as t to infinity, the difference between the solution and the approximation tends to zero in a norm that sharply picks out their difference in shape. More explicitly, the difference in shape between a solution and a quasi-static cnoidal-wave approximation is of the order of the damping rate times the norm of the cnoidal-wave at each instant.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
van Groesen, E.
Date : 1 September 1996
DOI : 10.1137/S003614109325342X
Uncontrolled Keywords : asymptotic behavior, cnoidal waves, perturbed KdV equation
Additional Information : Published in SIAM Journal on Mathematical Analysis, 27, 1424-1447. © 1996 Society for Industrial and Applied Mathematics.
Depositing User : Mr Adam Field
Date Deposited : 27 May 2010 14:40
Last Modified : 05 Jul 2019 16:29

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