Talk:PlanetPhysics/Soliton

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: soliton %%% Primary Category Code: 05.45.Yv %%% Filename: Soliton.tex %%% Version: 1 %%% Owner: invisiblerhino %%% Author(s): invisiblerhino %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in} \setlength{\headsep}{0.00in} \setlength{\headheight}{0.00in} \setlength{\evensidemargin}{0.00in} \setlength{\oddsidemargin}{0.00in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{9.00in} \setlength{\voffset}{0.00in} \setlength{\hoffset}{0.00in} \setlength{\marginparwidth}{0.00in} \setlength{\marginparsep}{0.00in} \setlength{\parindent}{0.00in} \setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners.

% almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts}

% used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\begin{document}

A soliton is a non-linear \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} which moves through space without dispersion at constant \htmladdnormallink{speed}{http://planetphysics.us/encyclopedia/Velocity.html}. They occur naturally as solutions to the Korteweg - de Vries equation. They were first observed by John Scott Russell in the 19th century and then by Martin Kruskal and Norman Zabusky (who coined the term soliton) in a famous \htmladdnormallink{computer simulation}{http://planetphysics.us/encyclopedia/ComputerSimulation.html} in 1965. Insight into solitons can be obtained by noting that the Korteweg - de Vries equation satisfies D'Alembert's solution: \[ u(x, t) = f(x-ct) + g(x+ct) \] We see at once that this satisfies two important criteria: it has a constant \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} $c$, and it can also be shown that the two \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} $f$ and $g$ can collide without altering shape. Solitons also occur in non-linear optics and as solutions to \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} equations in \htmladdnormallink{quantum field theory}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html}.

\end{document}