Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1996-05-22
Nonlinear Sciences
Exactly Solvable and Integrable Systems
15 pages LaTeX. The figures are available upon request (dorren@geof.ruu.nl)
Scientific paper
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.
Dorren H. J. S.
Snieder R. K.
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