DISSIPATION IN HAMILTONIAN-SYSTEMS - DECAYING CNOIDAL WAVES

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 .