Computer Algebra Recipes

308 CHAPTER 7. THE HUNT FOR SOLITONS
The lhs of SGE is simpli¯ed,
> check3:=simplify(lhs(SGE));
and the numerator of check3 expanded and further simpli¯ed.
> check4:=simplify(expand(numer(check3)));
check4 := 0
The result is zero, so U(x; t) satis¯es the sine{Gordon equation. This twosoliton
kink{kink solution is now animated, with c = 1
4 .
> animate(plot,[eval(U(x,t),c=1/4),x=-50..50],t=-100..100,
frames=50,thickness=2,axes=framed);
On running the animation, you will observe that the two kinks travel in opposite
directions, run into each other, and reverse directions after the collision, still
maintaining their initial shapes.
PROBLEMS:
Problem 7-16: Kink{antikink collision
In the two-soliton kink{kink solution, replace the ¯rst c by 1=c, x by ct,and
ct by x. Animate the resulting solution and show that it represents a kink{
antikink collision. Describe the observed behavior.
7.3 Simulating Soliton Collisions
Both authors of this text have spent an academic lifetime jousting with nonlinearities
in all mathematical shapes and sizes. In this text, we have tried to
provide a glimpse of the excitement and complexity involved in the study of
nonlinear dynamics, yet still present the bread-and-butter recipes necessary to
solve linear ODE and PDE problems, the staple of most undergraduate science
curricula. Whether the balance of linear and nonlinear recipes is right in our
computer algebra menu, you will have to be the judge, but we could not resist
presenting two numerical recipes that simulate soliton collisions. The ¯rst is
for the Korteweg{de Vries equation, the second for the sine{Gordon equation.
7.3.1 To Be or Not to Be a Soliton
There is no means of proving it is preferable to be than not to be.
E. M. Cioran, French philosopher (1911{1995)
To prove that solitary-wave solutions are solitons, i.e., whether they survive
collisions with each other unchanged in shape, is an important area of research
in nonlinear dynamics. One approach is to numerically collide the solitary
waves using a ¯nite di®erence scheme to simulate the relevant nonlinear PDE.
This was done by Norman Zabusky and Martin Kruskal [ZK65] for the KdV