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September 2, 2009

Where Have All the Solitons Gone?

Posted by David Corfield

I have been asked for suggested reading to help with a school project on solitons. Partial derivatives aren’t on the syllabus, so even a brief sketch of the balancing of dispersion and steepening of waves gleaned from Palais’s excellent survey is already stretching the bounds of knowledge. Rather than more mathematics then, something along the lines of physical and engineering applications would go down well.

There are some handy websites, e.g., here, which reminded me that a number of years ago there was a project to use solitons to send vast amounts of data rapidly down optical fibres. Erbium-doping was supposed to be the answer to transmission problems. But that was more than a decade ago. Are solitons actually used practically today in this way?

As for natural phenomena, it seems that Jupiter’s Great Red Spot is an autosoliton, and that solitons have been observed emerging from the Straits of Gibraltar.

Any other candidates?

Posted at September 2, 2009 9:40 AM UTC

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Re: Where Have All the Solitons Gone?

Possible replacement for Fitzhugh-Nagumo model ?

Soliton model

Posted by: John Yates on September 2, 2009 11:51 AM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

Thanks! It’s a small world. With a couple of links (via vector soliton) I’m brought to some of John’s future colleagues studying graphene in Singapore.

Here, use of atomic layer graphene as saturable absorber in a mode-locked fiber laser for the generation of ultrashort soliton pulses (756 fs) at the telecommunication band is demonstrated.

Posted by: David Corfield on September 2, 2009 12:35 PM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

A thought from left field…

If partial derivatives are not on the syllabus, this might be a bit much, but thought I would mention that it is fairly straightforward to build a (1-D) simulation of optical solitons. A great way to learn about something is to simulate it (remember the light mills?)

I once did a project (albeit in grad school) on solitons. I think it took 30-60 minutes to work out the equations AND write the code (with visualization), so it is not a massive under taking.

Posted by: Eric Forgy on September 2, 2009 3:16 PM | Permalink | Reply to this

hypothetical solitons in intensinal perostalsis; Re: Where Have All the Solitons Gone?

See (5) below. We’re still revising the submission to Nature, and the NIH grant application.

Excerpt from [Comprehensive Survey and Proposed Integrated Multidisciplinary Multiscale Complex Systems Model of Postsurgical Ileus (paralysis of small intestine) and normal function in adult human and various animal, in vivo, in vitro, and in silico literature, by Jonathan Vos Post and Thomas L. Vander Laan, M.D., F.A.C.S., FCCWS]

Our research on the Enteric Nervous System [the “second brain” of 10^11 neurons in your gut] suggests that there are 8 different myoelectrical dynamical behaviors of the human small intestine, based on mathematical modeling and simulations, including solitons (solitary waves). Some of these categories of dynamics have been clinically observed. Others remain to be seen. We speculate that this classification is clinically significant.

The 8 different system dynamics that we expect to observe in the simulation:

(1) no peristalsis, or amplitude too small to matter;
(2) single wave pulse propagating;
(3) wave train (which goes at different velocity from single wave pulse);
(4) alternans (every other wave in wave-train suppressed from subtle
2nd-order interaction) (first observed and widely published-about in Cardiac modeling and in vitro, but uses same Fitzhugh-Nagumo equations, cf. GIOME);
(5) soliton;
(6) chaos (as in tetrodotoxin experiment? [Tetrodotoxin = anhydrotetrodotoxin 4-epitetrodotoxin, tetrodonic acid, TTX, is a potent neurotoxin with no known antidote, which blocks action potentials in nerves by binding to the pores of the voltage-gated, fast sodium channels in nerve cell membranes]);
(7) reverse peristalsis;
(8) obstruction (in which peristalsis occurs only in a proper subset of the simulated gastrointestinal tract).
[there is also possibly a 9th solution to the system of differential equations, Canard, which I suspect is a mathematical artifact not likely to be physical, or, if it occurs, probably beneath the resolution of current observational capability].

Posted by: Jonathan Vos Post on September 2, 2009 4:14 PM | Permalink | Reply to this

Re: hypothetical solitons in intensinal perostalsis; Re: Where Have All the Solitons Gone?

Something’s fishy here, given that the cerebral cortex only has on the order of 10^10 neurons. Are you really asserting that the ENS has hundreds of billions of neurons, ten times as many as in the cerebrum?

Posted by: John Armstrong on September 2, 2009 5:36 PM | Permalink | Reply to this

Gut-feel for exponents; Re: hypothetical solitons in intensinal perostalsis; Re: Where Have All the Solitons Gone?

Sorry, my typo. I meant 10^9 neurons in your gut. I recently discussed this at length with Christoph Koch, at Caltech.

When I was in undergrad Bio (Caltech 1968-1973) the textbooks said that there were roughly 10^10 (ten billion) neurons in the human brain. When I was doing my M.S. and PhD work in grad school (1973-1977) the textbooks said: “Of the 10 billion neurons in your brain, 100 billion are Glial Cells.”

To give a popularized citation: 10^9 neurons (especially Interstitial Cells of Cajal) in the plexus distributed between the two muscle layers in your stomach, small intestine, and large intestine, that being 1% as many neurons as in your brain, and many more than your spinal cord? Some people are intellectuals but only at the gut level. “Our understanding of virtue and vice, success and failure, has long been expressed in the language
of appetite, consumption, and digestion” [‘A Short History of the American Stomach’ by Frederick Kaufman, Harcourt, 2008, 206 pp.]

I can email a draft paper related to this to anyone who wants to be distracted.

Posted by: Jonathan Vos Post on September 2, 2009 8:44 PM | Permalink | Reply to this

Dr. Michael D. Gershon, author of “The Second Brain”; Re: Gut-feel for exponents; Re: hypothetical solitons in intensinal perostalsis; Re: Where Have All the Solitons Gone?

The Other Brain Also Deals With Many Woes
By HARRIET BROWN
Published: August 23, 2005

Dr. Michael D. Gershon [author of “The Second Brain” and the chairman of the department of anatomy and cell biology at Columbia], “who coined the term ‘second brain’ in 1996, is one of a number of researchers who are studying brain-gut connections in the relatively new field of neurogastroenterology. New understandings of the way the second brain works, and the interactions between the two, are helping to treat disorders like constipation, ulcers and Hirschprung’s disease.”

“The role of the enteric nervous system is to manage every aspect of digestion, from the esophagus to the stomach, small intestine and colon. The second brain, or little brain, accomplishes all that with the same tools as the big brain, a sophisticated nearly self-contained network of neural circuitry, neurotransmitters and proteins.”

“The independence is a function of the enteric nervous system’s complexity.”

“‘Rather than Mother Nature’s trying to pack 100 million neurons someplace in the brain or spinal cord and then sending long connections to the GI tract, the circuitry is right next to the systems that require control,’ said Jackie D. Wood, professor of physiology, cell biology and internal medicine at Ohio State.”

“Two brains may seem like the stuff of science fiction, but they make literal and evolutionary sense.”

Posted by: Jonathan Vos Post on September 3, 2009 6:42 PM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

David wrote:

I have been asked for suggested reading to help with a school project on solitons.

What age group are we talking about here?

For youngsters, I recommend topological solitons. Take a length of ribbon and hang it stretched out between two walls with a twist in it. If you wiggle the ribbon a little and things are set up right, you can see the twist move along like a little “particle”:

I think I’ve also seen a twist cancel an “antitwist”. In theory, you should also be able to see a twist pass right through an antitwist:

or two twists bounce off each other:

But I’m not sure I’ve seen that happen in an actual ribbon. I think I’ve seen the ‘breather mode’, though:

For more, read about the sine-Gordon equation. This equation assumes there’s a force that makes the ribbon want to lie flat.

Posted by: John Baez on September 3, 2009 5:31 PM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

What age group are we talking about here?

17-18.

It seems one can convey something of the KdV equation. The dispersion due to components travelling with different velocities under

u t+u xxx=0, u_t + u_{x x x} = 0,

and steepening for

u t+uu x=0. u_t + u u_x = 0.

On the applied side of things, are sine-Gordon solitons supposed to relate to physical particles?

Posted by: David Corfield on September 3, 2009 9:48 PM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

The simple sine-Gordon model I was talking about is a model where ‘space’ (the ribbon) is 1-dimensional. It’s the simplest, most intuitively obvious theory with topological solitons.

You should really think of the sine-Gordon model as a theory where space is compactified to a circle and the field takes values in a circle. In other words, take a ribbon and attach the ends together. Then at any time the field is a continuous function

ϕ:S 1S 1 \phi : S^1 \to S^1

and the number of twists in your ribbon (usually called the ‘kink number’ in this game) is just the winding number. It’s a ‘topologically conserved charge’ — an integer that’s conserved for topological reasons. The word ‘charge’ is physics jargon here; if I were explaining this to kids I’d just call it the ‘number of particles’ or ‘number of twists’.

There are various generalizations; one of the most exciting is where space becomes 3-dimensional, and the field takes values in a 3-sphere. Again we compactify space, so at any time the field is a function

ϕ:S 3S 3 \phi : S^3 \to S^3

and the conserved charge is the ‘winding number’ arising from π 3(S 3)=\pi_3(S^3) = \mathbb{Z}.

Of course this works with any number, not just 1 or 3. But the fun thing about 3 is that we live in 3 dimensional space, S 3SU(2)S^3 \cong SU(2), and SU(2)SU(2) acts as a symmetry group in the simplest old-fashioned theory of protons, neutrons and pions! We now think of this SU(2)SU(2) as the approximate symmetry that acts on the space 2\mathbb{C}^2 whose basis vectors are the up and down quark. Protons, neutrons and pions are made of just these two quarks — the lightest quarks.

Anyway, there turns out to be a fairly successful — but phenomenological, not fundamental — model of protons, neutrons and pions in which the basic field at any moment in time is a map

ϕ:S 3S 3 \phi : S^3 \to S^3

It’s called the Skyrme model. The protons and neutrons are topological solitons in this model, just like twists in that ribbon!

So, you can tell the kids that there’s a theory where protons and neutrons are like twists in a ribbon… but try not to fool them into thinking this theory is ‘true’: it’s just pretty good for certain purposes.

Witten revived interest in the Skyrme model back when he was just starting to be Witten. He noticed that the fact π 4(S 3)=/2\pi_4(S^3) = \mathbb{Z}/2 can be used to explain why protons and neutrons are fermions: the process of switching two of them gives a map ϕ:S 4S 3\phi : S^4 \to S^3 that corresponds to the nontrivial element of /2\mathbb{Z}/2, and by adding a ‘Wess–Zumino–Witten term’ to the action of the Skyrme model, you can make this process give rise to a phase of 1-1 for purely topological reasons. Switching two fermions is supposed to give a phase of 1-1, you see.

Anyway, this is just the beginning of a long story, but probably more than enough for those students.

If you can dig up Witten’s original paper on this, it’s a lot clearer than anything I was easily able to find using Google. This paper probably dates back to the 70’s or 80’s, back when physicists were just starting to use homotopy groups.

Posted by: John Baez on September 3, 2009 10:17 PM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

” He noticed that the fact π 4(S 3)=ℤ/2 can be used to explain why protons and neutrons are fermions: the process of switching two of them gives a map ϕ:S 4→S 3 that corresponds to the nontrivial element of ℤ/2, and by adding a ‘Wess–Zumino–Witten term’ to the action of the Skyrme model, you can make this process give rise to a phase of −1 for purely topological reasons. Switching two fermions is supposed to give a phase of −1, you see.”

I didn’t read Witten’s paper, but maybe that becomes less mysterious if we notice that instead of ” But the fun thing about 3 is that we live in 3 dimensional space”, that the fun is that we live in 4d and QCD is 3d, the forms.

So, we have to find the symmetry when we get a 4d space, the parameters of the fields that maks space-time, and embed it into a 3d dimensional space, 3forms of the theory. That, the 4homotopy group of the 3 sphere.

Posted by: Daniel de Franša MTd2 on September 4, 2009 1:40 AM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

If you can dig up Witten’s original paper on this…

Perhaps one of paper 281-283 here:

281) ‘Global aspects of current algebra’, Nuclear Physics B, Volume 223, Issue 2, 22 August 1983, Pages 422-432.

282) ‘Current algebra, baryons and quark confinement’, Nuclear Physics B, Volume 223, Issue 2, 22 August 1983, Pages 433-444.

283) With Gregory S. Adkins and Chiara R. Nappi, ‘Static properties of nucleons in the Skyrme model’, Nuclear Physics B, Volume 228, Issue 3, 5 December 1983, Pages 552-566.

A glimpse of 281 and a glimpse of 282.

Posted by: David Corfield on September 4, 2009 11:48 AM | Permalink | Reply to this

Re: Where Have All the Solitons Gone?

I guess at this place I should advertize an interactive applet made by Ulrich Pinkall at:

Virtual Math Labs

Moreover If I remember correctly Antti Niemi and coworkers might have done some numerical simulations on a system related to the Skyrme model, which displays knotted solitons.

Posted by: nad on September 4, 2009 6:49 AM | Permalink | Reply to this

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