Computer Algebra Recipes

288 CHAPTER 7. THE HUNT FOR SOLITONS
of having soliton solutions? This chapter is about the hunt for solitary waves
(possible solitons), using graphical and analytic approaches, and the analytic
con¯rmation that some of these solitary waves are indeed solitons.
First, we should have some idea of what solitary waves look like, and what
well-known nonlinear PDEs of physical interest are known to have them. To
keep the discussion simple, let's restrict our attention to wave motion in one
spatial dimension. Three well-known nonlinear PDEs that describe di®erent
types of wave motion are the Korteweg{de Vries equation (KdVE), the sine{
Gordon equation (SGE), and the nonlinear SchrÄodinger equation (NLSE):
² @Ã
@t
@Ã
+ ®Ã
@x + @3Ã 3 =0, KdVE;
@x
² @2Ã 1
2 =
@x c2 ² i @Ã
@x
@ 2 Ã
2 +sinÃ, SGE;
@t
1 @
§
2
2 Ã
@t 2 + jÃj2 Ã =0, NLSE.
Here x is the spatial coordinate, t is the time, ® is a numerical scale parameter,
i = p ¡1, and Ã(x; t) is the amplitude. All three equations turn out to be very
important in nonlinear dynamics because, under suitable approximations, they
arise in many di®erent contexts.
The KdVE has been used to describe [SCM73] water waves in shallow canals,
magnetohydrodynamic waves in plasmas, longitudinal dispersive waves in elastic
rods, pressure waves in liquid{gas bubble mixtures, and so on.
The SGE is applicable [BEMS71] to the propagation of magnetic spins in
ferromagnets, magnetic °ux in Josephson junctions, crystal dislocations, ultrashort
optical pulses, etc.
Undoubtedly, the most important application of the NLSE [Has90] is to the
propagation of optical pulses in glass ¯bers whose refractive index n is of the
form n = n0 + n1 I, withn0 and n1 positive constants and I the light intensity.
The light intensity is proportional to jÃj 2 , where à is the complex electric
¯eld amplitude that satis¯es the NLSE. The light intensity is experimentally
measured rather than the electric ¯eld amplitude.
The derivation of these three nonlinear wave equations is beyond the scope
of this text. The reader who is interested in such matters should consult the
references cited above. As will be demonstrated, all three nonlinear PDEs
support collisionally stable solitary-wave solutions, i.e., solitons.
What do solitary waves (solitons) look like? Figure 7.1 shows a sketch of the
two commonly occurring types. For the peaked variety (called nontopological
solitary waves by mathematicians), there exists a localized maximum with the
pulse dropping to zero amplitude at §1. Nontopological solitary waves also
exist whose pulse amplitude displays a localized dip to zero with the pulse
increasing to a constant nonzero amplitude at §1. In terms of the intensity,
the NLSE supports both types of nontopological solitary waves. Those optical
solitary waves with peaks are called bright solitary waves, while those with the