A stability analysis for the Korteweg-de Vries equation
Abstract
In this paper the stability of the Korteweg-de Vries (KdV) equation is investigated. It is shown analytically and numerically that small perturbations of solutions of the KdV-equation introduce effects of dispersion, hence the perturbation propagates with a different velocity then the unperturbed solution. This effect is investigated analytically by formulating a differential equation for perturbations of solutions of the KdV-equation. This differential equation is solved generally using an Inverse Scattering Technique (IST) using the continuous part of the spectrum of the Schr\"{o}dinger equation. It is shown explicitly that the perturbation consist of two parts. The first part represents the time-evolution of the perturbation only. The second part represents the interaction between the perturbation and the unperturbed solution. It is shown explicitly that singular non-dispersive solutions of the KdV-equation are unstable.
Cite
@article{arxiv.solv-int/9605005,
title = {A stability analysis for the Korteweg-de Vries equation},
author = {H. J. S. Dorren and R. K. Snieder},
journal= {arXiv preprint arXiv:solv-int/9605005},
year = {2008}
}
Comments
15 pages LaTeX. The figures are available upon request (dorren@geof.ruu.nl)