Welcome to Klaus Brauer's SOLITON Page
One of the most exciting phenomena in dealing with non-linear Partial Differential Equations are the Solitons, i.e. solitary waves. The first person reporting these phenomena was the Scottish engineer John Scott Russel, who described the propagation of a wave in shallow water.
Nowadays we have better knowledge of the underlying mathematical properties. Solitons are the solutions of the famous non-linear Korteweg - de Vries Equation. A solution to this PDE may be found in using the method of Bäcklund transform.
Korteweg - de Vries Equation
The solution may be visualizied as a 3D Plot and as a Density Plot
(both generated with Mathematica 8.0.4).
Finally it can be nicely observed by looking at the animated graphs, produced as well
with Mathematica, Version 8 (available since 2011).
Analytical solution and graphical representation of the One Soliton solution.
It is possible to construct solutions to the Korteweg - de Vries equation which are non-linear superpositions of regular and
irregular single solutions.
Vvedensky, Dimitri D.
The author of this Web page has written an article (20 pages as a PDF file). The contents
points out to some history, presents Vvedensky's solutions, and shows some Mathematica
This construction method has been performed for two and for three superpositioned solutions. Each of them have a parameter, say β1 for
the one soliton solution resp. β1 and β2 for two waves or
β1, β2 and β3 for three waves.
The effect is that a wave travels the faster the greater that parameter is - thus overtaking a slower wave.
Update: April 10th, 2012Zurück zu Klaus Brauers Heimatseite         Back to Klaus Brauer's Homepage