Computer Algebra Recipes

310 CHAPTER 7. THE HUNT FOR SOLITONS
Using the operator F, we add the three time-dependent solitary waves in f, and
take the time derivative of f in g.
> f:= add(F(X[i],c[i]),i=1..3): g:=diff(f,t):
Setting the time to zero, we plot the input pro¯le over the spatial range x =0
to M, the resulting picture being shown in Figure 7.8.
> t:=0: plot(f,x=0..M,thickness=2);
2.5
2
1.5
1
0.5
0 50 100 x 150 200 250
Figure 7.8: Input pro¯le for the three-solitary-wave collision simulation.
Note that there is a slight overlap of the solitary-wave tails. One could place
them further apart initially, but this takes more computing time.
To evaluate f and g at the spatial mesh points, we use the unapply command
to turn them into operators in terms of the spatial coordinate x.
> f2:=unapply(f,x): g2:=unapply(g,x):
Using these two operators, we calculate the input amplitudes Ui;0 ´ f(i; t =0)
and Ui;1 ´ f(i; t =0)+kg(i; t =0)fori =0toM in the ¯rst and second
initial conditions, ic1 and ic2 .
> ic1:=seq(U(i,0)=evalf(f2(i)),i=0..M):
> ic2:=seq(U(i,1)=evalf(f2(i))+k*g2(i),i=0..M):
To avoid any possible unknown U values creeping into the double do loop that
will be used to iterate the numerical algorithm, we \initialize" all U values to
zero for i =0toM and j =2toN, i.e., for all remaining grid points. These
zeros will be overwritten as the loop is executed and new U values calculated.
> init:=seq(seq(U(i,j)=0,i=0..M),j=2..N):
The two initial conditions and the initialization are assigned.
> assign(ic1,ic2,init):