Computer Algebra Recipes

316 CHAPTER 7. THE HUNT FOR SOLITONS
The kink is on the left, the antikink on the right, the two pro¯les being joined
together in the middle. At the x boundaries, one has U ¼ 2 ¼ and @U=@x ¼ 0.
Since the kink is traveling to the right and the antikink to the left, a collision will
take place before each reaches the opposite boundary. Provided that we do not
let the kink and antikink come close to those boundaries, the above boundary
conditions will prevail for all times in the run. This implies that @U=@t ¼ 0at
the x boundaries. In the following do loop, the x-boundary grid points (circled
points in Figure 7.9) are initialized, setting U =2¼ and p = q =0.
> for j from 2 to N by 2 do #boundary conditions
> u[0,j]:=2*evalf(Pi): p[0,j]:=0: q[0,j]:=0;
> u[M,j]:=2*evalf(Pi): p[M,j]:=0: q[M,j]:=0;
> end do:
The following double do loop calculates the values of u at the other grid points.
> for j from 0 to (N-1) do:
A conditional statement is inserted to start i at i0 =0forj =0; 2; 4; ::: and
i0 =1forj =1; 3; 5; ::::
> if j mod 2 = 0 then i0:=0: else i0:=1: end if;
The following do loop runs over the spatial grid points i, incrementing them
from i0 to M ¡ 2instepsof2.
> for i from i0 to (M-2) by 2 do
Using equation (7.15), we calculate the values of p and q.
> p[i+1,j+1]:=0.5*(p[i+2,j]+p[i,j]+q[i+2,j]-q[i,j]
+(-sin(u[i+2,j])+sin(u[i,j]))*h);
> q[i+1,j+1]:=0.5*(p[i+2,j]-p[i,j]+q[i+2,j]+q[i,j]
+(-sin(u[i+2,j])-sin(u[i,j]))*h);
Then U is evaluated using equations (7.16) and (7.17), and the average taken.
> uP1:=u[i,j]+0.5*h*(p[i,j]+p[i+1,j+1] +q[i,j]+q[i+1,j+1]);
> uP2:=u[i+2,j]+0.5*h*(-p[i+2,j]-p[i+1,j+1]+q[i+2,j]
+q[i+1,j+1]);
> u[i+1,j+1]:=(uP1+uP2)/2;
> end do: end do:
The results are plotted,
> for j from 0 to N by 2 do
> pl(j):=plot([seq([xmin+2*i*h,u[2*i,j]],i=0..M/2)],
thickness=3,labels=["x","U"]):
> end do:
andanimatedwiththeinsequence=true option.
> plots[display]([seq(pl(2*j),j=0..N/2)],insequence=true);
When the work sheet is executed, the kink{antikink hump °ips upside down
but the shape of the moving pro¯le is identical to the input shape aside from a