Computer Algebra Recipes

7.3. SIMULATING SOLITON COLLISIONS 317
phase factor e i¼ = ¡1, indicating that the kink and antikink solitary waves are
indeed solitons. Finally, the CPU time is calculated.
> cpu:=(time()-begin)*seconds;
cpu := 3:024 seconds
and is about 3 seconds on a 3-GHz PC.
PROBLEMS:
Problem 7-21: Ampli¯ed kink{antikink input
In the text recipe, double the amplitudes of the input kink and antikink solitarywave
pro¯les. Remembering to also double the value 2¼ in the initialization
statement to avoid causing an end-e®ect problem, run the ¯le with the ampli¯ed
input and interpret the outcome.
Problem 7-22: Kink{kink collision
Modify the recipe to simulate the collision of a kink solitary wave with another
kink. Discuss the observed behavior.
Problem 7-23: Antikink{antikink collision
Modify the recipe to simulate the collision of an antikink solitary wave with
another antikink. Discuss the observed behavior.
Problem 7-24: Rectangular mesh
Solve the problem of the text recipe using an explicit scheme based on a rectangular
mesh.
Problem 7-25: Interacting laser beams
The interaction of two intense laser pulses of di®erent frequencies as they pass
through each other in opposite directions in a certain resonant absorbing °uid
can be described [RE76] by the following normalized PDEs for the laser intensities
U and V ,
@U @U
+
@x @y = ¡g1
@V @V
UV ¡ ®U; ¡
@x @y = ¡g2 UV + ®V:
Here x is the normalized distance inside the °uid medium of length one unit, y
the normalized time, g1 > 0andg2 > 0arethe\gain"coe±cients,and® ¸ 0
the absorption coe±cient. The U pulse travels in the positive x direction, while
the V pulse moves in the negative x direction.
(a) Find the characteristic directions along which the PDEs reduce to ODEs.
(b) Devise an explicit numerical scheme that integrates the ODEs along the
characteristic directions assuming that there are no pulses initially inside
the °uid (U(x; 0) = V (x; 0) = 0 for 0 < x < 1) and identical ¯niteduration
U and V pulses are fed in at opposite ends (U(0;y)=V (1;y)=
f(y) for0· y · Y = 1 and zero for y>Y).
2
(c) Numerically solve the equations and animate the results, assuming that
f(y) =1,g1 =0:4, g2 = 20, and (a) ® =0,(b)® =0:5.
(d) Discuss the behavior of the two pulses as revealed in the animation.