Dissipation in Hamiltonian systems: decaying cnoidal waves


Derks, G. and Groesen, E. van (1996) Dissipation in Hamiltonian systems: decaying cnoidal waves. SIAM Journal on Mathematical Analysis, 27 (5). pp. 1424-1447. ISSN 0036-1410

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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 \infty $, 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
Copyright:© 1996 SIAM
Electrical Engineering, Mathematics and Computer Science (EEMCS)
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Link to this item:http://purl.utwente.nl/publications/30277
Official URL:https://doi.org/10.1137/S003614109325342X
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