Computer Algebra Recipes

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dip are referred to as black solitary waves. The bright ones occur for the plus
sign in the NLSE, the black ones for the minus sign. The Korteweg{de Vries
(KdV) equation also possesses nontopological solitary-wave solutions.
− ∞
U
peaked kink
z
Figure 7.1: Qualitative shapes of two common types of solitary waves.
For the kink type (referred to as topological solitary waves) in Figure 7.1, the
pulse amplitude U changes from one constant value (e.g., zero in the ¯gure) at
¡1 to a larger constant value at + 1. The region in which the change takes
place is usually quite localized. Antikink solitary waves canalsoexist,forwhich
the amplitude changes from a constant value at ¡1 to a lower constant value
at + 1. The SGE displays both kink and antikink solutions.
Given a nonlinear wave equation, how are these solitary-wave solutions
found? We know that the one-dimensional linear wave equation has a general
solution of the structure Ã(x; t) =f(x ¡ ct)+g(x + ct), where f and g
are arbitrary functions. The function f(x ¡ ct) describes a waveform traveling
with speed c in the positive x-direction, while the form g(x + ct) describes a
wave traveling in the negative x-direction. Let us now con¯ne our attention to
waves traveling in the positive x-direction, the discussion for waves traveling
in the opposite direction being similar. The linear wave equation can support
localized solutions Ã(x; t) =f(z = x ¡ ct) such as the peaked solitary waves
shown in Figure 7.1. These solutions will translate unchanged in shape along
the positive x-axis. For nonlinear wave equations, we can look for similar types
of localized solutions. That is to say, we seek solutions of the mathematical form
Ã(x; t)=Ã(z = x ¡ ct) that have pro¯les qualitatively similar to those shown
in Figure 7.1. Note that this assumed form reduces the number of independent
variables from two (x and t) toone(z), thus reducing the PDE to an ODE.
Borrowing concepts from Chapter 1, we can make a phase-plane portrait for
the nonlinear ODE. As will be shown in the following section, the solitary-wave
solutions will correspond to separatrix solutions,aseparatrixbeingatrajectory
in the phase plane that divides the plane into regions with qualitatively di®erent
behaviors. The graphical method of hunting for solitons is quite important,
because it allows for their possible existence to be established, even if analytic
forms do not exist.
∞