Computer Algebra Recipes

312 CHAPTER 7. THE HUNT FOR SOLITONS
Problem 7-19: Ampli¯cation
Multiply the smallest of the three solitary waves in the text recipe by a factor of
3 and interpret the resulting behavior when the worksheet is executed. Explore
the e®ect of multiplying one or more pulses by numerical factors.
Problem 7-20: Radiative ripples
In the text recipe, change the sech 2 terms in the input pulses to sech 4 terms,
then execute the modi¯ed recipe, and discuss the results.
7.3.2 Are Diamonds a Kink's Best Friend?
I never hated a man enough to give him diamonds back.
Zsa Zsa Gabor, Movie actress (1919{)
Although the sine{Gordon equation,
@2Ã @x2 ¡ @2Ã =sinÃ;
@t2 (7.7)
can be numerically solved with an explicit scheme based on a rectangular mesh,
it lends itself more naturally to being tackled with a diamond-shaped mesh
chosen to follow the characteristic directions of the equation. To see how this
works, let's consider the general PDE
a @2U @x2 + b @2U @x@y + c @2U + e =0;
@y2 (7.8)
where a, b, c, and e are functions of U, @U=@x, @U=@y, but not of higher
derivatives. For the SGE, U = Ã, y = t, a =1,b =0,c = ¡1, and e = ¡ sin Ã.
Equation (7.8) can be quite generally solved by the method of characteristics.
Setting p ´ @U=@x and q ´ @U=@y, equation (7.8) can be written in the form
a @p @p @q
+ b + c + e =0:
@x @y @y
Since p = p(x; y) andq = q(x; y), then
(7.9)
dp @p @p dy
= + ;
dx @x @y dx
dq @q dx @q
= + :
dy @x dy @y
(7.10)
Substituting @p=@x and @q=@y into (7.9), then multiplying through by dy=dx,
noting that @q=@x = @p=@y, and rearranging yields
" μ 2 μ # ·
@p dy dy
a ¡ b + c ¡ a
@y dx dx
dp
¸
dy dq dy
+ c + e =0:
dx dx dx dx
(7.11)
At this stage, the resulting equation looks like a mathematical mess! However,
if we choose to work in the characteristic directions whose slopes are given by
μ 2 μ
dy dy
a ¡ b + c =0; (7.12)
dx dx