Computer Algebra Recipes

7.1. THE GRAPHICAL HUNT FOR SOLITONS 297
@Á1 @Á1
+ v1
@t @x = ¡¯1 Á2 Á3;
@Á2 @Á2
+ v2
@t @x = ¯2 Á1 Á3;
@Á3 @Á3
+ v3
@t @x = ¡¯3 Á1 Á2:
(7.1)
Here, x is the direction of wave propagation, t is the time, and the coupling
parameters ¯1, ¯2, and¯3 are real and positive. Our goal is to show graphically
that there exists a set of three solitary waves, one for each equation, which will
propagate along together at a common velocity c.
The DEtools and PDEtools library packages are loaded,
> restart: with(DEtools): with(PDEtools):
and the parameter °, which will shortly be introduced, is unprotected from its
Maple assignment as Euler's constant. To generate the N = 3 PDEs
> unprotect(gamma): N:=3:
with a do loop, let's set Á4 = Á1 and Á5 = Á2.
> phi[4](x,t):=phi[1](x,t): phi[5](x,t):=phi[2](x,t):
The following do loop then generates the three relevant PDEs.
> for j from 1 to N do
> pde[j]:=diff(phi[j](x,t),t)+v[j]*diff(phi[j](x,t),x)
=(-1)^j*beta[j]*phi[j+1](x,t)*phi[j+2](x,t);
> end do;
³ ´ ³ ´
pde 1 := @@t Á1(x; t) + @ v1 Á1(x; t) = ¡¯1 Á2(x; t) Á3(x; t)
@x
³ ´ ³ ´
pde 2 := @@t Á2(x; t) + @ v2 Á2(x; t) = ¯2 Á3(x; t) Á1(x; t)
@x
³ ´ ³ ´
pde 3 := @@t Á3(x; t) + @ v3 Á3(x; t) = ¡¯3 Á1(x; t) Á2(x; t)
@x
Solitary-wave solutions are sought that are functions of the single \new" independent
\spatial" variable z = x ¡ ct, c being an arbitrary velocity for the
moment. The relevant variable transformation is entered, with t = ¿ and new
amplitudes U1(z), U2(z), and U3(z).
> tr:=fx=z+c*tau,t=tau,phi[1](x,t)=U[1](z),
phi[2](x,t)=U[2](z),phi[3](x,t)=U[3](z)g;
tr := fx = z+c¿; t= ¿; Á1(x; t) =U1(z); Á2(x; t) =U2(z); Á3(x; t) =U3(z)g
and the dchange command applied to each PDE in the following do loop.
> for j from 1 to N do
> ode[j]:=dchange(tr,pde[j],[z,tau,U[1](z),U[2](z),U[3](z)]);
To simplify the output, the substitution ¯j = °j (vj ¡ c) ismadeinthejth
ODE and each equation divided by (vj ¡ c) and simpli¯ed.
> ode[j]:=subs(beta[j]=gamma[j]*(v[j]-c),ode[j]):