On the simultaneous effect of viscosity and variable depth on the propagation of solitary waves

The Korteweg-de Vries equation modified by both the effect of viscosity and the effect of variable depth is derived and the evolution of a solitary wave in the presence of both of them is discussed by the method of multiple scales. The analysis has been focused on the eventual balance between both effects, which might allow a solitary wave to preserve its initial shape. It has been shown that cither the amplitude or the length or the speed of the wave can only be preserved and the corresponding forms of the channel have been found.