Clas Blomberg - Physics of life-Elsevier Science (2007)

284 Part VII. Non-linearity
This means that we then get a first-order stationary solution of the original eq. (27.11):
qx ( ) j ⋅ sin( px) j sin( npx)
1
∞
∑
n
n2
(27.18)
with j-values given by as in eqs. (27.15) and (27.17).
If b 0 in the basic eq. (27.11), the second-order terms disappear and we get in lowest order:
j
1
8
3c a
( D p2
)
(27.19)
j n is in this case then in lowest order proportional to j 3 1.
This can be regarded as a standard method to investigate the appearance of spatially
inhomogeneous solutions. Often, as above, the lowest mode (j 1 ) dominates, but this is not
always the case, in particular not in higher dimensions.
27E
Non-linear waves
Next we consider some features of non-linear waves. A wave here means that it is some spatial
structure that moves with a certain wave velocity. There are some noticeable distinctions
between these waves and more traditional linear waves. For non-linear waves, there is no
superposition principle: Two waves do not sum up to a common wave pattern with interference
phenomena. Non-linear waves do not show interference. Non-linear oscillations are
more rigid than linear, harmonic oscillations. They are often very stable and more insensitive
to disturbances than linear waves. The interaction between separate non-linear waves is completely
different to that of linear waves. In particular, it is possible that non-linear waves interact
as ordinary particles: they can collide, and get out from the collision as colliding particles
with a final state determined by the conservation of momentum and energy. Waves with such
a particle-like behaviour are called solitons. They constitute a particular a particular type of
the non-linear waves, and are the most studied group. They can also annihilate each other.
Non-linear waves usually form particular patterns, which are completely determined by the
basic equations. This means that the amplitude and the velocity are directly related (this is not
the case for a linear wave). Although one may analyse a wave by a mode description, it is not
as a linear wave composed by constituting components. A more general concept than soliton
is a solitary wave: a single wave pattern. To establish the soliton character, one shall besides
the wave behaviour, consider the interaction features, which may be a difficult task.
The solitons are much devoted studies in recent times. Besides their obvious importance
in, for instance, plasma physics, a clear reason for this is that there exist very elegant analytic
methods for their study. These studies are mainly confined to a small number of equations,
which are studied in much detail.
It may be appropriate here to say some words of the background. Early inspirations to
the mathematical studies came from direct observations and detailed descriptions of solitary,