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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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Abstract

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
Additional Information: Published in SIAM Journal on Mathematical Analysis, 27, 1424-1447. © 1996 Society for Industrial and Applied Mathematics.
Uncontrolled Keywords: asymptotic behavior, cnoidal waves, perturbed KdV equation
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/1408

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