### 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 language | English |
---|---|

Pages (from-to) | 1424-1447 |

Number of pages | 24 |

Journal | SIAM journal on mathematical analysis |

Volume | 27 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1996 |

### Keywords

- Perturbed KdV equation
- Cnoidal waves
- Asymptotic behavior

## Fingerprint Dive into the research topics of 'Dissipation in Hamiltonian systems: decaying cnoidal waves'. Together they form a unique fingerprint.

## Cite this

Derks, G., & van Groesen, E. (1996). Dissipation in Hamiltonian systems: decaying cnoidal waves.

*SIAM journal on mathematical analysis*,*27*(5), 1424-1447. https://doi.org/10.1137/S003614109325342X