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\noindent {\bf Problem statement} a) {\bf The $1$-dimensional wave
equation} Suppose that $H(x,t)$ represents the height of a vibrating
string over the point $x$ of the real line at time $t$. Then for small
vibrations and for homogeneous strings (think of a guitar string) $H$
satisfies the partial differential equation ${{\partial^2 H
}\over{\partial x^2}}-{{\partial^2 H }\over{\partial t^2}}=0$. Verify
that if $f(w)$ is any twice-differentiable function of one variable,
then $H(x,t)=f(x-t)$ satisfies the wave equation. Comment on how the
wave shape (the graph of $f$) travels along the string as $t$
changes. If $H_1$ and $H_2$ satisfy the wave equation, verify that
$H_1+H_2$ and $cH_1$ (where $c$ is any constant) also satisfy the wave
equation. This is called ``the principle of superposition''.

\medskip

\noindent b) {\bf The Korteweg-de Vries equation} The following
paragraph was written in 1844 by John Scott Russell, a Scottish
engineer.

\font\eightrm=cmr8
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%$u_t+u_{xxx}+6uu_x=0$

%http://www.ma.hw.ac.uk/solitons/press.html press release 

%http://www.ma.hw.ac.uk/~chris/scott\_russell.html

{\parindent=.2in \q \narrower

\noindent I was observing the motion of a boat which was rapidly drawn
along a narrow channel by a pair of horses, when the boat suddenly
stopped -- not so the mass of water in the channel which it had put in
motion; it accumulated round the prow of the vessel in a state of
violent agitation, then suddenly leaving it behind, rolled forward
with great velocity, assuming the form of a large solitary elevation,
a rounded, smooth and well-defined heap of water, which continued its
course along the channel apparently without change of form or
diminution of speed. I followed it on horseback, and overtook it still
rolling on at a rate of some eight or nine miles an hour, preserving
its original figure some thirty feet long and a foot to a foot and a
half in height. Its height gradually diminished, and after a chase of
one or two miles I lost it in the windings of the channel. Such, in
the month of August 1834, was my first chance interview with that
singular and beautiful phenomenon which I have called the Wave of
Translation.

} 

\medskip

\noindent This quotation is from {\tt
http://www.ma.hw.ac.uk/$\tilde{\hphantom{e}}$chris/scott\_russell.html}.
See also the {\tt Wikipedia} article about John Scott Russell.

\medskip

This was the first published record of what is now called a {\it
soliton}. In 1895 the phenomenon was described with a partial
differential equation named the {\it Korteweg-de Vries equation}
(KdV): $u_t+u_{xxx}+6uu_x=0$ (here $u(x,t)$ is the vertical
displacement of the wave at time $t$ and position $x$). Korteweg and
de Vries stated the equation with supporting reasoning. This equation
is non-linear and does not satisfy the ``principle of
superposition''. KdV and related equations have turned out to be very
important both theoretically and in practical applications (fiber
optics, transmission of nerve impulses, some chemical reactions). The
following exercises are probably more appropriately done with the help
of a tool such as {\tt Maple}.

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\noindent i) If $K>0$, show that $u_K={K\over 2}\left({\rm sech}
\!\left({{\sqrt{K}}\over 2}(x-Kt)\right)\right)^{\! 2}$ is a solution of
KdV. Here ``sech'' means {\it hyperbolic
secant}.

\medskip

\noindent ii) Verify that $7u_1$ and $u_2+u_3$ are {\it not}\/
solutions of KdV.

\medskip

\noindent iii) Describe a connection between the speed of the soliton
and its maximum height. (In 1885, Russell wrote
``The sound of a cannon travels faster than the command to
fire it.'')

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\noindent {\bf More information} {\tt
http://math.cofc.edu/faculty/kasman/SOLITONPICS}











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