Solitons Strike Back

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Solitons Strike Back

• Brendan DuBree
• Chrissy Maher
• Angela Piccione

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• Previously, we discussed solitons which are
  stable, non-linear solitary waves which behave
  like a particle and neither change shape nor
  velocity. John Scott-Russell first discovered the
  soliton phenomenon in 1834, and further research
  led to understanding solitons as solutions to the
  KdV, mKdV, and Sine-Gordon equations. When two
  solitons collide, they merge into one and then
  separate into two with the same shape and
  velovity as before the collision. Solitons are
  used in physics, electronics, optics, technology,
  and biology.

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Shallow Water Waves - KdV

• General KdV Equation ut uxxx auux 0
• most fundamental equation for solitons
• Has soliton solutions for one-directional shallow
  water waves in a rectangular canal
• Two-Soliton solution of the KdV equation

u
72 3 4cosh(2x 8t) cosh(4x 64t)
a 3cosh(x 28t) cosh(3x 36t)2
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Shallow Water Waves - KP

• 2D generalization of KdV KP Equation
• (ut 6uux uxxx)x 3uyy 0
• subscripts denote partial derivatives
• setting a 6 from KdV
• Two-soliton solution
• u(x, y, t) 2?2ln(1 ef1 ef2 A12ef1f2) /
  ?x2
• fi kix liy ?it are phase variables
• (ki, li) are the wave vectors
• ?i are the frequencies
• A12 is the phase shift parameter

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distant pacific storms produce nearly perfect KdV
soliton waves that travel from a reef about 1 mi
off the coast of Molokai, Hawaii 1
interaction of two solitons of unequal amplitudes
2
interaction of two soliton waves in shallow ocean
water off the coast of Oregon 3
interaction of soliton-like surface waves in very
shallow water on Lake Peipsi, Estonia in July
2003 2
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Solitons on a Molecular Level

• Proteins complex molecules of carbon, hydrogen,
  nitrogen, and oxygen
• Perform key functions of cells
• grab molecules and assemble them into cellular
  structures
• tear molecules apart for energy
• transport oxygen and other necessary items from
  one cell to another

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• Proteins perform these function in cells by
  jerking, stretching, flipping, and twisting into
  whatever shapes are required for the job
• Biologists understanding of how proteins
  function is a lot like your and my understanding
  of how a car works. We know you put in gas and
  the gas is burned to make things turn but the
  details are all pretty vague. (Alwyn Scott in
  Discover Magazine, Vol. 15 No. 12, Dec. 1994)

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• According to traditional thought, a burst of
  energy would distort a protein but scatter
  through the protein in a trillionth of a second,
  like dropping a rock into a puddle
• 1970s A. S. Davydov suggested that solitons
  occur in this energy transfer
• Myosin has long sections consisting primarily of
  a chain of pairs of carbon and oxygen atoms
• Davydov proposed that a wave traveling along such
  a chain would experience a compressing effect
• This could balance the dispersing tendency
  VOILA!! SOLITON!

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Concerns with Davydovs Model

• Its hard (impossible?) to observe actual
  proteins at work
• It applied mathematics from a 1D theory to 3D
  proteins
• Are solitons stable at biologically relevant
  temperatures?
• Most studies conducted at absolute zero
• 1985 experiments conducted at 300K showed that
  Davydov solitons lasted for only a few
  picoseconds, and so couldnt explain energy
  transfer
• 1994 counter-arguments using quantum mechanics
  suggest that Davydov solitons may have a longer
  lifespan
• Moral we still dont know how proteins transfer
  energy, but Davydov solitons could be a possible
  explanation

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Typhoons as Solitons

• A typhoon is a 3D cyclone vortex with a warm,
  low-pressure center, formed over tropical oceans
• It acquires helical structure under the action of
  Coriolis force due to the earths rotation
• Typhoons are mainly affected by 3 factors
• Dispersion makes the wave shape wider
• Dissipation decreases the wave amplitude
• Advection steepens the convex wave shape

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Typhoons as Solitons

• When the 3 factors are in equilibrium, they drive
  a typhoon forward with stable structure and
  constant speed
• Four scientists did an experiment in which they
  simulated two typhoons in a glass enclosure using
  air, cigarette smoke, and heaters, and watched
  them collide.
• After the 2 typhoons collided, they separated and
  restored their respective shapes and velocities
• These properties make typhoons seem like big, 3D
  solitons

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These are pictures from the scientists
experiment 4
We can see the typhoons collide, mix, and then
separate again.
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Solitons in Space

• Empty Space isnt really empty there could be
  pockets of energy which spring up and then shrink
  as the energy flows out to lower-energy space
  around them
• Friedberg and Lee asked what would happen if
  quarks appeared inside a shrinking higher energy
  pocket of space
• The shrinking is a compressing effect
• Quarks repel when they get too close - dispersing
  effect
• The result would be a soliton consisting of
  unbound quarks trapped inside the bubble
• These soliton bubbles could be as big as several
  light years across, the size and mass of a
  million billion (1,000,000,000,000,000,000,000)
  suns

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Solitons in Space

• These soliton stars could explain two big
  scientific puzzles
• There is energy streaming out of galaxies, which
  many astrophysicists attribute to giant black
  holes. But soliton stars might make more
  mathematical sense.
• They could account for dark matter, which
  possibly provides 90 of the universes mass but
  is undetectable by normal means.
• Problem as in the molecular case, observation in
  nature is hard
• Do these solitons exist and explain many
  scientific phenomenon? We dont know. But they
  could.

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References

• 1 The KP Page. http//www.amath.washington.edu/
  bernard/kp.html
• 2 Soomere and Engelbrecht. Extreme Elevations
  and Slopes of Interacting Kadomtsev-Petviashvilli
  Solitons in Shallow Water.
• 3 Physics Today, Vol. 44 Issue 3, March 1991
• 4 Songnian, et. al. Rotating Annulus
  Experiment Large-Scale Helical Soliton in the
  Atmosphere. Physical Review E, Vol. 64, Dec.
  2000
• 5 Infeld et. al. Decay of Kadomtsev-Petviashvil
  i Solitons. Physical Review Letters. Vol. 72 No.
  9, Feb. 1994
• 6 Freedman, David. Lone Wave. Discover
  Magazine, Vol. 15 No. 12, Dec. 1994
• 7 Cruzeiro-Hansson. Two Reasons Why the
  Davydov Solution May Be Thermally Stable After
  All. Physical Review Letters, Vol. 73 No. 21,
  Nov. 1994
• 8 Lombdalh, P.S. and W. C. Kerr. Do Davydov
  Solitons Exist at 300K? Physical Review Letters,
  Vol. 55 No. 11, Sept. 1985