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The Korteweg-de Vries Equation: History, exact Solutions, and graphical Representation

Travelling waves as solutions to the Korteweg-de Vries equation (KdV) which is a non-linear Partial Differential Equation (PDE) of third order have been of some interest already since 150 years. The author’s aim is to present an analytical exact result to the KdV equation by means of elementary operations as well as by using Backlund transform. Special interest is devoted to non-linear superposition of several waves, the so called Solitary Waves or Solitons which is performed by Backlund transform too. The derivation of the exact solutions follows the presentation of Vvedenski (1992). It is fascinating that these exact solutions which involve a lot of calculations can be done with the help of the Computer Algebra System Mathematica © (see Wolfram (1999), not only presenting analytical expressions but in addition 3D-Plots, Contour Plots, and 2D-Plots for discrete values of time, finally animating these 2D-Plots and presenting these plots as well as the solution formulae at an Internet page. Another aim of the author has been to apply numerical discrete methods to non-linear PDEs. With the help of the exact solutions gained here an excellent possibility is given to test the quality of the numerical methods by checking the numerical results against the exact solution.