Computer Algebra Recipes

7.3. SIMULATING SOLITON COLLISIONS 315
The spatial range is taken from xmin =3x1 = ¡15 to xmax =3x2 = 15, and
the step size h =(xmax ¡ xmin)=M calculated.
> xmin:=3*x[1]: xmax:=3*x[2]: h:=evalf((xmax-xmin)/M);
h := 0:1500000000
The kink{antikink pro¯le at time t is entered.
> U:=4*arctan(exp((x-x[1]-c[1]*t)/a))
+4*arctan(exp(-(x-x[2]-c[2]*t)/b));
³
U := 4 arctan e (1:666666667 x +8:333333335 ¡ 1:333333334 t)´
³
+ 4 arctan e (¡1:666666667 x +8:333333335 ¡ 1:333333334 t)´
The values of p(x; 0) ´ (@U=@x)jt=0 and q(x; 0) ´ (@U=@t)jt=0 are needed, so
the relevant spatial and time derivatives are calculated and labeled P and Q.
> P:=diff(U,x): Q:=diff(U,t): t:=0:
Setting t = 0, the following loop evaluates U, p, andq at each spatial grid point
on the zeroth time step. Note that the numerical value of U at the ith spatial
grid point is labeled ui;0 and i is incremented in steps of 2 from 0 to M.
> for i from 0 to M by 2 do #initial conditions
> x:=xmin + i*h;
> u[i,0]:=evalf(U); p[i,0]:=evalf(P); q[i,0]:=evalf(Q);
> end do:
The input pro¯le is now plotted and displayed in Figure 7.10.
> plot([seq([xmin+2*i*h,u[2*i,0]],i=0..M/2)],view=
[xmin..xmax,-1..15],tickmarks=[3,3],labels=["x","U"]);
15
10
U
5
–10 0
x 10
Figure 7.10: Input pro¯le for the kink{antikink soliton collision simulation.