TY - JOUR
T1 - Dissipation in Hamiltonian systems
T2 - decaying cnoidal waves
AU - Derks, G.
AU - van Groesen, E.
PY - 1996
Y1 - 1996
N2 - 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 → ∞, 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.
AB - 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 → ∞, 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.
KW - Perturbed KdV equation
KW - Cnoidal waves
KW - Asymptotic behavior
U2 - 10.1137/S003614109325342X
DO - 10.1137/S003614109325342X
M3 - Article
SN - 0036-1410
VL - 27
SP - 1424
EP - 1447
JO - SIAM journal on mathematical analysis
JF - SIAM journal on mathematical analysis
IS - 5
ER -