Computer Algebra Recipes

7.1. THE GRAPHICAL HUNT FOR SOLITONS 293
Felix Bloch. Under the in°uence of an applied magnetic ¯eld, the Bloch wall
(kink soliton) can propagate according to the SGE without changing in shape.
PROBLEMS:
Problem 7-1: Variation in velocity
Explore how the sine{Gordon solitary waves vary in shape as the velocity c is
altered.
Problem 7-2: Cosine{Gordon equation
If the sine term is replaced with a cosine in the SGE, how would the solitarywave
solutions be a®ected? Con¯rm your reasoning by running the text recipe
with a cosine present, instead of the sine term. You will have to alter the initial
conditions to obtain the new separatrixes.
Problem 7-3: Is there or isn't there?
Suppose that the nonlinear term in the SGE is replaced with sin 2 Ã. Using
the phase-plane portrait approach, determine whether there is a solitary-wave
solution to this modi¯ed SGE.
7.1.2 In Search of Bright Solitons
We're all of us sentenced to solitary con¯nement inside our
own skins, for life!
Tennessee Williams, American dramatist (1914{1983)
Our second example illustrates the existence of a bright solitary-wave solution
to the NLSE for the situation that the equation has the plus sign.
The DEtools library package is loaded,
> restart: with(DEtools):
and the NLSE entered.
> NLSE:=I*diff(E(x,t),x)+(1/2)*diff(E(x,t),t,t)
+abs(E(x,t))^2*E(x,t)=0;
μ
@
NLSE := E (x; t) I +
@x 1
μ
2 @
E(x; t) + jE (x; t)j
2 @t2 2 E (x; t) =0
Because of the complex nature of the equation, a slightly di®erent assumption
is made here than in the sine{Gordon example. In this case, a solitary-wave
solution of NLSE of the form E(x; t) =U(t) e ibx is sought, where the parameter
b, the coordinate x, andU(t) are taken as positive.
> ode:=eval(NLSE,E(x,t)=U(t)*exp(I*b*x))
assuming b>0,x>0,U(t)>0;
ode := ¡U (t) be (bxI) + 1
μ
2 d
U (t) e
2 dt2 (bxI) + U (t) 3 e (bxI) =0
This assumption has reduced the NLSE to an ODE, which is simpli¯ed by
dividing by eibx and multiplying by 2.