G. Derks et E. Vangroesen, DISSIPATION IN HAMILTONIAN-SYSTEMS - DECAYING CNOIDAL WAVES, SIAM journal on mathematical analysis, 27(5), 1996, pp. 1424-1447
The uniformly damped Korteweg-de Vries (KdV) equation with periodic bo
undary conditions can be viewed as a Hamiltonian system with dissipati
on added. The KdV equation is the Hamiltonitan part and it has a two-d
imensional family of relative equilibria. These relative equilibria ar
e space-periodic soliton-like waves, known as cnoidal waves. Solutions
of the dissipative system, starting near a cnoidal wave, are approxim
ated with a long curve on the family of cnoidal waves. This approximat
ion curve consists of a quasi-static succession of cnoidal waves. The
approximation process is sharp in the sense that as a solution tends t
o zero as t --> infinity, the difference between the solution and the
approximation tends to zero in a norm that sharply picks out their dif
ference in shape. More explicitly, the difference in shape between a s
olution 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
.