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Solitons Strike Back

Solitons Strike Back. Brendan DuBree Chrissy Maher Angela Piccione.

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Solitons Strike Back

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  1. Solitons Strike Back Brendan DuBree Chrissy Maher Angela Piccione

  2. 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.

  3. 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

  4. 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

  5. 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]

  6. 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

  7. 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)

  8. 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!

  9. 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

  10. 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

  11. 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

  12. These are pictures from the scientists’ experiment [4] We can see the typhoons collide, mix, and then separate again.

  13. 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

  14. 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.

  15. 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

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