Solitons Strike BackSolitons Strike Back

Brendan DuBreeChrissy 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 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.

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.

Shallow Water Waves - KdVShallow Water Waves - KdV• General KdV Equation: uGeneral KdV Equation: utt + u + uxxxxxx + + ααuuuuxx = 0 = 0

– most fundamental equation for solitonsmost fundamental equation for solitons

• Has soliton solutions for one-directional Has soliton solutions for one-directional shallow water waves in a rectangular canalshallow water waves in a rectangular canal

• Two-Soliton solution of the KdV equation:Two-Soliton solution of the KdV equation:

u =u =72 3 + 4cosh(272 3 + 4cosh(2xx – 8 – 8tt) + cosh(4) + cosh(4xx – 64 – 64tt))

αα [3cosh( [3cosh(xx – 28 – 28tt) + cosh(3) + cosh(3xx – 36 – 36tt)])]22

Shallow Water Waves - KPShallow Water Waves - KP

• 2D generalization of KdV: KP Equation2D generalization of KdV: KP Equation

(u(utt + + 66uuuuxx + u + uxxxxxx))xx + 3u + 3uyyyy = 0 = 0– subscripts denote partial derivativessubscripts denote partial derivatives– setting setting αα = 6 from KdV = 6 from KdV

• Two-soliton solution:Two-soliton solution:u(x, y, t) = 2u(x, y, t) = 2∂∂22ln(1 + eln(1 + eφφ11 + e + eφφ22 + A + A1212eeφφ1+1+φφ22) / ) /

∂∂xx22

– φφii = k = kiix + lx + liiy + y + ωωiit are phase variablest are phase variables– (k(kii, l, lii) are the wave vectors) are the wave vectors– ωωii are the frequencies are the frequencies– AA1212 is the phase shift parameter is the phase shift parameter

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 soliton-like surface

waves in very shallow water on Lake

Peipsi, Estonia in July 2003 [2]

interaction of two soliton

waves in shallow ocean water

off the coast of Oregon [3]

Solitons on a Molecular LevelSolitons on a Molecular Level

Proteins: complex molecules of carbon, Proteins: complex molecules of carbon, hydrogen, nitrogen, and oxygenhydrogen, nitrogen, and oxygen

Perform key functions of cells:Perform key functions of cells: grab molecules and assemble them into grab molecules and assemble them into

cellular structurescellular structures tear molecules apart for energytear molecules apart for energy transport oxygen and other necessary items transport oxygen and other necessary items

from one cell to anotherfrom one cell to another

Proteins perform these function in cells by Proteins perform these function in cells by “jerking, stretching, flipping, and twisting “jerking, stretching, flipping, and twisting into whatever shapes are required for the into whatever shapes are required for the job”job”

““Biologists’ understanding of how proteins Biologists’ understanding of how proteins function is a lot like your and my function is a lot like your and my understanding of how a car works. We understanding of how a car works. We know you put in gas and the gas is burned know you put in gas and the gas is burned to make things turn but the details are all to make things turn but the details are all pretty vague.” (Alwyn Scott in pretty vague.” (Alwyn Scott in Discover Discover MagazineMagazine, Vol. 15 No. 12, Dec. 1994), Vol. 15 No. 12, Dec. 1994)

According to traditional thought, a burst of According to traditional thought, a burst of energy would distort a protein but scatter energy would distort a protein but scatter through the protein in a trillionth of a second, like through the protein in a trillionth of a second, like dropping a rock into a puddledropping a rock into a puddle

1970s: A. S. Davydov suggested that solitons 1970s: A. S. Davydov suggested that solitons occur in this energy transferoccur in this energy transfer

Myosin has long sections consisting primarily of Myosin has long sections consisting primarily of a chain of pairs of carbon and oxygen atomsa chain of pairs of carbon and oxygen atoms

Davydov proposed that a wave traveling along Davydov proposed that a wave traveling along such a chain would experience a compressing such a chain would experience a compressing effecteffect

This could balance the dispersing tendency … This could balance the dispersing tendency … VOILA!! SOLITON!VOILA!! SOLITON!

Concerns with Davydov’s ModelConcerns with Davydov’s ModelIt’s hard (impossible?) to observe actual proteins at It’s hard (impossible?) to observe actual proteins at workworkIt applied mathematics from a 1D theory to 3D proteinsIt applied mathematics from a 1D theory to 3D proteinsAre solitons stable at biologically relevant Are solitons stable at biologically relevant temperatures?temperatures? Most studies conducted at absolute zeroMost studies conducted at absolute zero 1985: experiments conducted at 300K showed that 1985: experiments conducted at 300K showed that

Davydov solitons lasted for only a few picoseconds, and Davydov solitons lasted for only a few picoseconds, and so couldn’t explain energy transferso couldn’t explain energy transfer

1994: counter-arguments using quantum mechanics 1994: counter-arguments using quantum mechanics suggest that Davydov solitons may have a longer lifespansuggest that Davydov solitons may have a longer lifespan

Moral: we still don’t know how proteins transfer energy, Moral: we still don’t know how proteins transfer energy, but Davydov solitons could be a possible explanationbut 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 These are pictures from the scientists’ from the scientists’

experiment [4]experiment [4]

We can see the We can see the typhoons collide, mix, typhoons collide, mix,

and then separate again.and then separate again.

Solitons in SpaceSolitons 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

• 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.

• 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.

ReferencesReferences[1][1] The KP Page.The KP Page.

http://www.amath.washington.edu/~bernard/kp.htmlhttp://www.amath.washington.edu/~bernard/kp.html[2][2] Soomere and Engelbrecht. “Extreme Elevations and Soomere and Engelbrecht. “Extreme Elevations and

Slopes of Interacting Kadomtsev-Petviashvilli Solitons in Slopes of Interacting Kadomtsev-Petviashvilli Solitons in Shallow Water.”Shallow Water.”

[3][3] Physics TodayPhysics Today, Vol. 44 Issue 3, March 1991, Vol. 44 Issue 3, March 1991[4][4] Songnian, et. al. “Rotating Annulus Experiment: Songnian, et. al. “Rotating Annulus Experiment:

Large-Scale Helical Soliton in the Atmosphere.” Large-Scale Helical Soliton in the Atmosphere.” Physical Physical Review EReview E, Vol. 64, Dec. 2000, Vol. 64, Dec. 2000

[5][5] Infeld et. al. “Decay of Kadomtsev-Petviashvili Infeld et. al. “Decay of Kadomtsev-Petviashvili Solitons.” Solitons.” Physical Review LettersPhysical Review Letters. Vol. 72 No. 9, Feb. . Vol. 72 No. 9, Feb. 19941994

[6][6] Freedman, David. “Lone Wave.” Freedman, David. “Lone Wave.” Discover MagazineDiscover Magazine, , Vol. 15 No. 12, Dec. 1994Vol. 15 No. 12, Dec. 1994

[7][7] Cruzeiro-Hansson. “Two Reasons Why the Davydov Cruzeiro-Hansson. “Two Reasons Why the Davydov Solution May Be Thermally Stable After All.” Solution May Be Thermally Stable After All.” Physical Physical Review LettersReview Letters, Vol. 73 No. 21, Nov. 1994, Vol. 73 No. 21, Nov. 1994

[8][8] Lombdalh, P.S. and W. C. Kerr. “Do Davydov Solitons Lombdalh, P.S. and W. C. Kerr. “Do Davydov Solitons Exist at 300K?” Exist at 300K?” Physical Review LettersPhysical Review Letters, Vol. 55 No. 11, , Vol. 55 No. 11, Sept. 1985Sept. 1985