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Klaus Brauer: On Solitons
Klaus Brauer: On Solitons (If no image appears below, "Click" your browser "Refresh" icon.) This post is in memoriam: John Scott Russell (Russel) [18081888] The soliton, which is the irreducible physical manifestation, that is the crux of Pulsoid Theory (PT). It is the soliton, “s,” that is heuristically symbolized by the Natural function, Psi, Ψ, x² – x (NF). It is the soliton that connects the sinusoidal equations of Light and the ellipsoidal equations of gravity. The soliton is a physical manifestation in which “likes” attract; thus, from a single manifestation evolves all the variety of existence. It was this problem of likes attracting (coalescence) that Philip Morrison helped resolve in 1955 by quickly referring to the 1871 publication of C. A . Bjerknes concerning hydrodynamical forces at a distance, as described and demonstrated, in detail, by his son, Vilhelm, which is referred to in the 1952 writings of Ludwig Prandtl. See: HeriotWatt University, Department of Mathematics for the history of the rediscovery of the soliton in 1973. Authors note: All the following information about solitons, on this page, is courtesy of: www.usf.uniosnabrueck.de/~kbrauer/solitons.html, and, Klaus Brauer’s Home Page. Note: The above links are often updated and are more current than the below quotes therefrom. For the serious reader the above links are recommended. ________________________________________ Welcome to Klaus Brauer's SOLITON Page ________________________________________ One of the most exciting phenomena in dealing with nonlinear Partial Differential Equations are the Solitons, i.e. solitary waves. The first person reporting these phenomena was the Scottish engineer John Scott Russell, 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 nonlinear Korteweg  de Vries Equation. A solution to this PDE may be found in using the method of Bäcklund transform. Korteweg  de Vries Equations (If no image appears below, "Click" your browser "Refresh" icon.) The solution may be visualized as a 3D Plot and as a Contour Plot (both generated with Mathematica 4.0). Finally it can be nicely observed by looking at the animated graph, produced as well with Mathematica 4.0. Analytical solution and graphical representation of the One Soliton solution. (If no image appears below, "Click" your browser "Refresh" icon.) It is possible to construct solutions to the Korteweg  de Vries equation which are nonlinear 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," AddisonWesley Publishing Company, Reading, MA, ISBN 0201544091, 1993 The author of this Web page has written an article (16 pages as a PDF file). The contents points out to some history, presents Vvedensky's solutions, and shows some Mathematica code. Watch the paper here with Acrobat Reader, Size: 1031 KB Download a ZIP version of the PDF file here, Size: 329 KB This construction method has been performed for two and for three superpositioned solutions. Each of them have a parameter, say b1 and b2 for two waves and b1, b2 and b3 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 Russell's experiments are given by HeriotWatt University (Edinburgh/Scotland). See: HeriotWatt University, Department of Mathematics The University of Kyoto/Japan has prepared a SolitonLab Art Gallery. Further Information, especially on the SineGordonEquation, 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. Concerning waves in general, please visit Waves, Waves, Waves. An easy introduction comes from the Technical University of Denmark Even a German TV channel has brought solitons to the public. Real solitons observed in the Strait of Gibraltar. ________________________________________ Update: August 17th, 2004 Zurück zu Klaus Brauers Heimatseite Back to Klaus Brauer's Homepage ________________________________________ (If no image appears below, "Click" your browser "Refresh" icon.) ©Copyright 20052008 by Brunardot. All rights reserved. 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