Dissipation in Hamiltonian systems: decaying cnoidal waves

G. Derks, E. van Groesen

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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 → ∞, 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.
    Original languageEnglish
    Pages (from-to)1424-1447
    Number of pages24
    JournalSIAM journal on mathematical analysis
    Volume27
    Issue number5
    DOIs
    Publication statusPublished - 1996

    Keywords

    • Perturbed KdV equation
    • Cnoidal waves
    • Asymptotic behavior

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