Computer Algebra Recipes

292 CHAPTER 7. THE HUNT FOR SOLITONS
the solutions are periodic as the trajectories cycle around vortex points at (U =
§ ¼; Y =0)asz increases. The solutions outside the separatrixes are also
oscillatory, resembling \over-the-top" motion for the undamped pendulum.
What do the separatrix solutions look like? Qualitatively, the answer is quite
simple. A trajectory starting at the saddle point (0,0) at z = ¡1 asymptotically
approaches the right saddle point (2 ¼; 0) as z ! + 1. A similar trajectory
connects the left saddle point at (¡2 ¼; 0) to the one at the origin as z varies
from ¡1 to + 1. These are examples of kink solitary waves, whose pro¯les
U(z) may be seen by entering F(z,U), thus producing Figure 7.3. The curves
connecting U =2¼ to U =0,andU =0to¡2 ¼, are the antikink solutions.
> F(z,U);
U
6
4
2
0
–2
–4
–6
2 4 6 8 10
z
Figure 7.3: Pro¯les of kink and antikink solitary waves.
An important physical example of a kink is a so-called Bloch wall between
two magnetic domains in a ferromagnet as schematically depicted in Figure 7.4.
The magnetic spins rotate from, say, spin down in one domain to spin up in
Figure 7.4: Bloch wall between two ferromagnetic domains.
the adjacent domain. The narrow transition region between down and up spins
is called a Bloch wall in honor of the theoretical physicist and Nobel laureate