Transcript Solitons Strike Back Brendan DuBree Chrissy Maher Angela Piccione Previously, we discussed solitons which are stable, non-linear solitary waves which behave like a particle and.

Solitons Strike Back
Brendan DuBree
Chrissy Maher
Angela Piccione
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
research
phenomenon
led
to
in
1834,
understanding
and
further
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.
Shallow Water Waves - KdV
• General KdV Equation: ut + uxxx + αuux = 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)
α
[3cosh(x – 28t) + cosh(3x – 36t)]2
Shallow Water Waves - KP
• 2D generalization of KdV: KP Equation
(ut + 6uux + uxxx)x + 3uyy = 0
– subscripts denote partial derivatives
– setting α = 6 from KdV
• Two-soliton solution:
–
–
–
–
u(x, y, t) = 2∂2ln(1 + eφ1 + eφ2 + A12eφ1+φ2) / ∂x2
φi = kix + liy + ωit are phase variables
(ki, li) are the wave vectors
ωi are the frequencies
A12 is the phase shift parameter
distant pacific storms produce
nearly perfect KdV soliton waves
interaction of two solitons of
that travel from a reef about 1 mi off
unequal amplitudes [2]
the coast of Molokai, Hawaii [1]
interaction of soliton-like surface
interaction of two soliton
waves in very shallow water on Lake
waves in shallow ocean water
Peipsi, Estonia in July 2003 [2]
off the coast of Oregon [3]
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
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)
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!
Concerns with Davydov’s Model
It’s 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 couldn’t explain energy transfer
1994: counter-arguments using quantum mechanics
suggest that Davydov solitons may have a longer lifespan
Moral: we still don’t know how proteins transfer
energy, but Davydov solitons could be a possible
explanation
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 earth’s 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
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
These are pictures
from the scientists’
experiment [4]
We can see the
typhoons collide, mix,
and then separate again.
Solitons in Space
• Empty Space isn’t 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
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 universe’s 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 don’t know. But they could.
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-Petviashvili 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