Some nonlinear partial differential equations have solitary wave, or soliton solutions. Solitons have a finite ``extension" and they ``survive" collisions with other solitons. These concepts are well illustrated by the sine-Gordon equation:
where 
is a variable depending on the position x and time t,
and m
is a parameter. The simplest physical system represented by this equation is
an array of pendula. Electronic excitations in polymers like (CH) 
,
magnetic chains, and other systems are also modeled by this equation.
(a) Show that the ``single-soliton",
is a solution to Eq. 5.12. Calculate 
in terms of v and
. Plot the solution for v=0. Show that is
close to 0 or 
for almost every x, except in a range of width
and estimate for v=0. Calculate the energy 
stored in the
system, relative to the energy of the 
solution.
Figure 5.1: Solution of Eq. 5.12 as expressed in Eq. 5.14
for a=4 and a=0.0021 and m=1.
(b) Show that the ``two-soliton" (or ``breather"),
is also a solution. Calculate . Calculate the energy and show that
it is always less than 
.
(c) In Figure 5.1 the two-soliton solution is plotted for two values of m. Demonstrate that, for small a and for most values of the time t, the solution can be approximated by the sum of two, appropriately selected ``one-soliton" solutions.
Laszlo Mihaly
Thu Oct 31 13:23:11 EST 1996
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