  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.
