Introduction to Soliton

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Introduction to Soliton

• Chapter 14 Nonlinear PDEs
• Section 2 Solitons
• Partial Differential Equation
• An Introduction
• By Walter A. Strauss

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Contents

• Soliton demonstration
• Direct Scattering Problem
• Inverse Scattering Problem

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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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Other soliton equations

• Sine-Gordon equation
• Nonlinear Schrödinger equation
• Kadomstev Petviashvili (KP) equation

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Kink solution of Sine-Gordon Equation
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Sine-Gordon EquationOne-Soliton Solution

• kink solution
• antiKink solution

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Sine-Gordon EquationTwo-Soliton Solution

• Kink-kink Collision
• Kink-antikink Collision
• Standing Breather
• Large amplitude Breather
• Small amplitude Breather

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Sine-Gordon EquationThree-Soliton Solution

• Standing Kink and Moving Breather Collision
• Standing Breather and Moving Kink Collsion

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Sine-Gordon pulson
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Part 2

• Direct Scattering Problem

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Part 3

• Inverse Scattering Problem