Ondas dispersivas – Korteweg – De – Vries (KDV)

Abstract

The present investigation it shows the detail of the calculations obtained for a part of the deduction of one of the most important equations of the current importance in the Mathematics, the equation of the Dutch scientists Diederick Johannes Korteweg and Gustav De Vries (KdV), call like that in honor to these out-standing mathematicians. This deduction was published in the year 1895 by the magazine " Philosophical Magazine and Journal of Science " in London; the same one that with this work they were coming to formalize a fact that had been revealed some years before by a mathematical model of water waves on slightly deep surfaces known as Solitons. The Solitons have as principal characteristics of which they are waves of great extent which speed of spread depends on his extent in contraposition to the linear waves, are dispersives but in turn they preserve his form, are waves of type solitary wave, interact between them or with finite obstacles in such a way that after the interaction they recover totally his properties before the interaction except phase changes. In the initial part of the realized investigation, one presents a historical introduction on the origin of this physical phenomenon happened in a channel of Scotland for Scott Russell; then there deduces the variation of the level of the water with regard to the time, which originates a differential partial equation that will allow to find the solution of the same one, to demonstrate this formula it is begun by the supposition of which the speeds horizontal and vertical and of the fluid they can be expressed by rapidly convergent series. Finally, the solution of the equation decides in the straight line for stationary waves, for this there is realized the analysis of different scenes that might be considered in the equation KdV, in addition for this equation there are used elliptical functions that can be seen as a generalization of the trigonometrical known functions.