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
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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.
The interested reader is referred to the book:
Vvedensky, Dimitri D.
Partial Differential Equations with Mathematica - Chapter 9
Addison-Wesley Publishing Company, Reading, MA, ISBN 0-201-54409-1, 1993
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
code.
Read the revised version (March 2012) of this article here with Acrobat Reader, Size: 767 KB
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.
The two or the three waves preserve their shapes even after the overtaking process.
Analytical solution and graphical representation of the Two Solitons solution
Analytical solution and graphical representation of the Three Solitons solution
Further Information:
- A lot of information including the revival of John Scott Russel's experiments are given by
Heriot-Watt University (Edinburgh/Scotland).
- The University of Kyoto/Japan has prepared a
Soliton-Lab Art Gallery.
- Further Information, especially on the Sine-Gordon-Equation, solved by using the
Computer Algebra System Maple is coming from Tver State University in Russia
(by the way: Tver is a partner city of Osnabrück), look at the page of
Andrey E. Miroshnichenko .
- Lots of Links may be found from
R. Victor Jones' Soliton Page.
- Even a
German TV channel has brought solitons to the public.
- Real solitons observed in
the Strait of Gibraltar.
Update: April 10th, 2012
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