Computer Algebra Recipes

294 CHAPTER 7. THE HUNT FOR SOLITONS
> ode2:=simplify(2*ode/exp(I*b*x));
μ
2 d
ode2 := ¡2 U (t) b + U (t) +2U (t)
dt2 3 =0
The second-order ODE is cast into two ¯rst-order ODEs by setting dU=dt = Y
in ode3 , and substituting this expression into ode2 .
> ode3:=diff(U(t),t)=Y(t); ode4:=subs(ode3,ode2);
ode3 := d
U (t) =Y(t)
dt
μ
d
ode4 := ¡2 U (t) b + Y (t) +2U (t)
dt 3 =0
Two phase-plane trajectories will be plotted for initial conditions very close to
the origin. The origin will be revealed to be a saddle point.
> ic:=[[U(0)=.01,Y(0)=0],[U(0)=-.01,Y(0)=0]]:
Taking b =1,anoperatorFis formed to apply the phaseportrait command
to ode3 and ode4 for speci¯ed scene parameters A and B.
> F:=(A,B)->phaseportrait([ode3,eval(ode4,b=1)],[U(t),Y(t)],
t=0..9,ic,scene=[A,B],U=-2..2,Y=-2..2,stepsize=0.05,
dirgrid=[20,20],color=red,linecolor=blue,arrows=MEDIUM):
The phase-plane portrait results on entering F(U,Y),
> F(U,Y);
the result being shown in Figure 7.5.
Y
2
1
–2 –1 0 1 2
U
–1
–2
Figure 7.5: Separatrixes correspond to bright solitary waves.
The arrows indicate the direction of increasing t, while the solid curves to the
left and right of the origin are the two separatrixes. There are vortex points at
(U = § 1, Y =0)andasaddlepointattheorigin. Theseparatrixlinetothe
right of the origin starts at U =0 for t !¡1,growstoamaximumpositive