📄 soliton.mht
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<H1 style=3D"MARGIN-BOTTOM: 0px">Soliton
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<TD style=3D"PADDING-TOP: 6pt" vAlign=3Dtop noWrap=20
width=3D"100%"> Also found in: <A=20
=
href=3D"http://www.thefreedictionary.com/Soliton">Dictionary/thesaurus</A=
>,=20
<A=20
=
href=3D"http://medical-dictionary.thefreedictionary.com/Soliton">Medical<=
/A>,=20
<A=20
=
href=3D"http://encyclopedia2.thefreedictionary.com/Soliton">Encyclopedia<=
/A>,=20
<A =
href=3D"http://encyclopedia.farlex.com/Soliton">Hutchinson</A></TD>
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<DIV class=3Ddablink>For the company of the same name, see =
.</DIV><BR><BR>In=20
<A class=3Dtip onmouseover=3Dt_i(1) onmouseout=3Dt_o(1)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/mathematics">mathematic=
s</A>=20
and <A class=3Dtip onmouseover=3Dt_i(2) onmouseout=3Dt_o(2)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/physics">physics</A>, =
a=20
<B>soliton</B> is a self-reinforcing solitary wave (a wave packet =
or=20
pulse) that maintains its shape while it travels at constant =
speed;=20
solitons are caused by a cancelation of <A class=3Dtip =
onmouseover=3Dt_i(3)=20
onmouseout=3Dt_o(3)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/nonlinearity">nonlinear=
</A>=20
and dispersive effects in the medium. ("Dispersive effects" refer =
to a=20
relationship between the frequency and the speed of waves in the =
medium.)=20
Solitons are found in many physical phenomena, as they arise as =
the=20
solutions of a widespread class of weakly nonlinear dispersive <A=20
class=3Dtip onmouseover=3Dt_i(4) onmouseout=3Dt_o(4)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/partial+differential+eq=
uation">partial=20
differential equations</A> describing physical systems. The =
soliton=20
phenomenon was first described by <A class=3Dtip =
onmouseover=3Dt_i(5)=20
onmouseout=3Dt_o(5)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/John+Scott+Russell">Joh=
n=20
Scott Russell</A> (1808=E2=80=931882) who observed a solitary wave =
in the <A=20
class=3Dtip onmouseover=3Dt_i(6) onmouseout=3Dt_o(6)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/Union+Canal+(Scotland)"=
>Union=20
Canal</A> (a canal in Scotland), reproduced the phenomenon in a =
wave tank,=20
and named it the "Wave of Translation".=20
<H2>Definition</H2>A single definition of a soliton is difficult =
to=20
procure. Drazin and Johnson (1989) ascribe 3 properties to =
solitons:=20
<OL>
<LI>They are of permanent form;=20
<LI>They are localised within a region;=20
<LI>They can interact with other solitons,and emerge from the =
collision=20
unchanged, except for a <A class=3Dtip onmouseover=3Dt_i(7)=20
onmouseout=3Dt_o(7)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/Phase+(waves)">phase=20
shift</A>. </LI></OL><BR><BR>More formal definitions exist, but =
they=20
require substantial mathematics. On the other hand, some =
scientists use=20
the term <EM>soliton</EM> for phenomena that do not quite have =
these three=20
properties (for instance, the 'light bullets' of <A class=3Dtip=20
onmouseover=3Dt_i(8) onmouseout=3Dt_o(8)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/nonlinear+optics">nonli=
near=20
optics</A> are often called solitons despite losing energy during=20
interaction).=20
<H2>Explanation</H2>To see how dispersion and non-linearity can =
interact=20
to produce permanent and localized wave forms, consider a pulse of =
light=20
traveling in glass. This pulse can be thought of as consisting of =
light of=20
several different frequencies; since glass shows dispersion, these =
different frequencies will travel at different speeds and the =
shape of the=20
pulse will therefore change over time. However, there is also the=20
non-linear <A class=3Dtip onmouseover=3Dt_i(9) onmouseout=3Dt_o(9) =
=
href=3D"http://encyclopedia.thefreedictionary.com/Kerr+effect">Kerr=20
effect</A>: the speed of light of a given frequency depends on the =
light's=20
amplitude or strength. If the pulse has just the right shape, the =
Kerr=20
effect will exactly cancel the effect of dispersion, and the =
pulse's shape=20
won't change over time: a soliton. See <A class=3Dtip =
onmouseover=3Dt_i(10)=20
onmouseout=3Dt_o(10)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/soliton+(optics)">solit=
on=20
(optics)</A> for a much more detailed description. <BR><BR>Many <A =
class=3Dtip onmouseover=3Dt_i(11) onmouseout=3Dt_o(11)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/exactly+solvable+model"=
>exactly=20
solvable models</A> have soliton solutions, including the =
Korteweg-de=20
Vries equation, the <A class=3Dtip onmouseover=3Dt_i(12) =
onmouseout=3Dt_o(12)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/nonlinear+Schr%F6dinger=
+equation">nonlinear=20
Schr=C3=B6dinger equation</A>, the coupled nonlinear =
Schr=C3=B6dinger equation, and=20
the <A class=3Dtip onmouseover=3Dt_i(13) onmouseout=3Dt_o(13)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/sine-Gordon+equation">s=
ine-Gordon=20
equation</A>. The soliton solutions are typically obtained by =
means of the=20
<A class=3Dtip onmouseover=3Dt_i(14) onmouseout=3Dt_o(14)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/inverse+scattering+tran=
sform">inverse=20
scattering transform</A> and owe their stability to the <A =
class=3Dtip=20
onmouseover=3Dt_i(15) onmouseout=3Dt_o(15)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/integrability">integrab=
ility</A>=20
of the field equations. The mathematical theory of these equations =
is a=20
broad and very active field of mathematical research. <BR><BR>Some =
types=20
of <A class=3Dtip onmouseover=3Dt_i(16) onmouseout=3Dt_o(16)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/tidal+bore">tidal=20
bore</A>, a wave phenomenon of a few rivers including the <A =
class=3Dtip=20
onmouseover=3Dt_i(17) onmouseout=3Dt_o(17)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/River+Severn">River=20
Severn</A>, are 'undular': a wavefront followed by a train of =
solitons.=20
Other solitons occur as the undersea <A class=3Dtip =
onmouseover=3Dt_i(18)=20
onmouseout=3Dt_o(18)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/internal+wave">internal=
=20
waves</A>, initiated by seabed <A class=3Dtip =
onmouseover=3Dt_i(19)=20
onmouseout=3Dt_o(19)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/topography">topography<=
/A>,=20
that propagate on the oceanic <A class=3Dtip onmouseover=3Dt_i(20) =
onmouseout=3Dt_o(20)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/pycnocline">pycnocline<=
/A>.=20
Atmospheric solitons also exist, such as the <A class=3Dtip=20
onmouseover=3Dt_i(21) onmouseout=3Dt_o(21)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/Morning+glory+cloud">Mo=
rning=20
Glory Cloud</A> of the <A class=3Dtip onmouseover=3Dt_i(22) =
onmouseout=3Dt_o(22)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/Gulf+of+Carpentaria">Gu=
lf=20
of Carpentaria</A>, where pressure solitons travelling in a <A =
class=3Dtip=20
onmouseover=3Dt_i(23) onmouseout=3Dt_o(23)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/temperature+inversion">=
temperature=20
inversion</A> layer produce vast linear <A class=3Dtip =
onmouseover=3Dt_i(24)=20
onmouseout=3Dt_o(24)=20
href=3D"http://encyclopedia.thefreedictionary.com/roll+cloud">roll =
clouds</A>. The recent and not widely accepted <A class=3Dtip=20
onmouseover=3Dt_i(25) onmouseout=3Dt_o(25)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/soliton+model">soliton =
model</A> in <A class=3Dtip onmouseover=3Dt_i(26) =
onmouseout=3Dt_o(26)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/neuroscience">neuroscie=
nce</A>=20
proposes to explain the signal conduction within <A class=3Dtip=20
onmouseover=3Dt_i(27) onmouseout=3Dt_o(27)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/neuron">neurons</A> as =
pressure solitons. <BR><BR>A <A class=3Dtip onmouseover=3Dt_i(28)=20
onmouseout=3Dt_o(28)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/topological+soliton">to=
pological=20
soliton</A>, or <A class=3Dtip onmouseover=3Dt_i(29) =
onmouseout=3Dt_o(29)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/topological+defect">top=
ological=20
defect</A>, is any solution of a set of <A class=3Dtip =
onmouseover=3Dt_i(30)=20
onmouseout=3Dt_o(30)=20
=
href=3D"http://encyclopedia.thefreedictionary.com/partial+differential+eq=
uation">partial=20
differential equations</A> that is stable against decay to the =
"trivial=20
solution" due to topological constraints, rather than due to the =
<A=20
=
href=3D"http://encyclopedia.thefreedictionary.com/integrability">integrab=
ility</A>=20
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