View source for Linear and nonlinear waves

===Solitary waves and [[soliton|solitons]]===

The correct term for a wave which is localized and retains its form
over a long period of time is: ''solitary wave''. However, a ''soliton''
is a solitary wave having the additional property that other solitons
can pass through it without changing its shape. But, in the literature
it is customary to refer to the solitary wave as a soliton, although
this is strictly incorrect [[#Tao-08|[Tao-08]]].

<span id="Fig9.10"></span>[[Image:twoSolitonKdV_Fig914.png|none|thumb|250px|right|Figure 15: Evolution of a two-soliton solution of the KdV equation. Image illustrates the collision of two solitons that are both moving from left to right. The faster (taller) soliton overtakes the slower (shorter) soliton.]]
Solitons are stable, nonlinear pulses which exhibit a fine balance
between non-linearity and dispersion. They often result from real
physical phenomena that can be described by PDEs that are ''completely  integrable'', i.e. they can be solved exactly. Such PDEs describe:
shallow water waves, nonlinear optics, electrical network pulses,
and many other applications that arise in mathematical physics. Where
multiple solitons moving at different velocities occur within the
same domain, collisions can take place with the unexpected phenomenon
that, first they combine, then the faster soliton emerges to proceed
on its way. Both solitons then continue to proceed in the the same
direction and eventually reach a situation where their speeds and
shapes are unchanged. Thus, we have a situation where a faster soliton
can overtake a slower soliton. There are two effects that distinguishes
this phenomena from that which occurs in a linear wave system. The
first is that the maximum height of the combined solitons is not equal
to the sum of the individual soliton heights. The second is that,
following the collision, there is a phase shift between the two solitons,
i.e. the linear trajectory of each soliton before and after the collision
is seen to be shifted horizontally - see figure ([[#Fig9.10|15]]).

Some additional discussion is given in section ([[#The Korteweg-de Vries equation|The Korteweg-de Vries equation]]) and
detailed technical overviews of the subject can be found in the works
by Ablowitz & Clarkson [[#Abl-91|[Abl-91]]], Drazin & Johnson [[#Dra-89|[Dra-89]]] and Johnson [[#Joh-97|[Joh-97]]].
Soliton theory is still an active area of research and a discussion
on the various types of soliton solution that are known is given by
Gerdjikov & Kaup [[#Ger-05|[Ger-05]]].

====Soliton types====

Soliton types generally fall into thee types:
  
* '''Humps (pulses)''' - These are the classic bell-shaped curves  that are typically associated with soliton phenomena.  
* '''Kinks''' - These are solitons characterized by either a ''monotonic  positive shift'' (kink) or a ''monotonic negative shift'' (anti-kink)  where the change in value occurs gradually in the shape of an ''s-type''  curve.  
* '''Breathers (bions)''' - These can be either stationary or travelling  ''soliton humps that oscillate'': becoming positive, negative,  positive and so on.  

More details may be found in Drazin and Johnson [[#Dra-89|[Dra-89]]].