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In the Name of Allah The Most Merciful
Beneficent
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REPRESENTED BY JABRAN RASHID REG
10-MS-MT-24 MS MATHEMATICS SUBMITTE
D TO Dr.Syed Tauseef Mohyud-Din
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Part 1

• Soliton Demonstration

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What is soliton?

• A soliton is a localized wave solution of a
  nonlinear PDE which is remarkably stable
• One PDE that has such a solution is the
  Korteweg-deVries (KdV) equation

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Solitary Wave one soliton
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Properties one soliton

• Behaves like a particle
• Travels with constant shape and velocity
• Animation

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Solitary Wave 2 solitons

• When two solitary waves get closer, they
  gradually deform
• Finally merge into a single wave packet
• This packet soon splits into two solitary waves
  with the same shape and velocity before
  "collision".

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Solitary Wave 2 solitons

• Two solitons travel to the right with different
  speeds and shapes

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Animation
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Waves collision 2 solitons
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Waves Collision Multi-soliton

• Animation
• The general formula for multi-soliton solution of
  the KdV equation is

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Who discovered soliton?

• John Scott Russell (1808-1882)

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John Scott Russel

• In 1834, while conducting experiments to
  determine the most efficient design for canal
  boats, John Scott Russell made a remarkable
  scientific discovery. As he described it in his
  "Report on Waves" (Report of the fourteenth
  meeting of the British Association for the
  Advancement of Science, York, September 1844
  (London 1845), pp 311-390, Plates XLVII-LVII).

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Russells report

• I was observing the motion of a boat which was
  rapidly drawn along a narrow channel by a pair of
  horses, when the boat suddenly stopped - not so
  the mass of water in the channel which it had put
  in motion it accumulated round the prow of the
  vessel in a state of violent agitation, then
  suddenly leaving it behind, rolled forward with
  great velocity, assuming the form of a large
  solitary elevation, a rounded, smooth and
  well-defined heap of water, which continued its
  course along the channel apparently without
  change of form or diminution of speed.

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Russells report

• I followed it on horseback, and overtook it
  still rolling on at a rate of some eight or nine
  miles an hour, preserving its original figure
  some thirty feet long and a foot to a foot and a
  half in height. Its height gradually diminished,
  and after a chase of one or two miles I lost it
  in the windings of the channel. Such, in the
  month of August 1834, was my first chance
  interview with that singular and beautiful
  phenomenon which I have called the Wave of
  Translation.

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Scott Russell Aqueduct

• 89.3m long
• 4.13m wide
• 1.52m deep
• On the union Canal
• Near Heroit-Watt Univ.

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Scott Russell Aqueduct
Solitary wave
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Scott Russell Aqueduct
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Soliton Machine

• A system for producing John Scott Russell's wave
  in Snibston Discovery Park in England
• The system has weights and pulleys for pulling
  different hull designs along a series of troughs

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Soliton Machine (continued)
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Soliton Machine (continued)
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Soliton Machine (continued)
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KdV solitary wave solution

• Korteweb and de Vries (1895) discovered the
  equation possesses the solitary wave solution
• KdV equation
• Traveling wave solution

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What is convection?

• "Convection" has several, related meanings in
  weather....but it always involves rising air. It
  usually refers to "moist convection", where the
  excess water vapor in rising air parcels
  condenses to form a cloud.

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The heat released through this condensation can
help to sustain the convection by warming the air
further and making it rise still higher, which
causes more water vapor to condense, so the
process feeds on itself.
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But convection can also be dry, as occurs on a
sunny day over the desert, or in more humid
regions early in the day before the convection
has become strong enough to form clouds. The sun
warms the ground, and convective air currents
help to remove the excess heat from the surface.
Dry convection also occurs during the day even
when clouds are not forming...you just can't see
it.
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Scattering Scattering is a general physical
process where some forms of radiation, such as
light, sound, or moving particles, are forced to
deviate from a straight trajectory by one or more
localized non-uniformities in the medium through
which they pass
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Separation of colors by a prism is an example of
dispersion
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The numerical method has been applied to flnd new
solitons, for example, recently Han and Xu used
the numerical approach to flnd new soliton
solutions for the generalized KdV equations it
appears that these solitons do not possess the
\closed-form" representations...
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