Our Memorial Article for Norman J. Zabusky (1929–2018)

Along with David Campbell and Alan Newell, I have written a memorial article for my collaborator Norman Zabusky, who died in February. It was posted online earlier today.

Playing "Soliton" in "Scrabble" (Lexulous)

Take a look at what I just played in “Scrabble” (Lexulous).

An ASCII Soliton Collision

I promised this to some of my Ph.D. students, so I drew it, and I'm posting it here as well.

Tales from the ArXiv: "Sheep Soliton"

You had me at "sheep soliton".

Yes, under a suitable approximation, it appears that you can gave a soliton description of certain types of collective behavior in sheep.

Below are the title and abstract. (I was hoping for an intriguing picture from the experimental data, but alas the visuals in this article are run-of-the-mill.) In other contexts, when sheep (a type of 'active matter') behave like a fluid, there are obvious jokes to tell about "shearing instabilities".

Title: Sheep Soliton

Abstract: Monitoring small groups of sheep in spontaneous evolution in the field, we decipher behavioral rules that sheep follow at the individual scale in order to sustain collective motion. Individuals alternate grazing mode at null speed and moving mode at walking speed, so cohesive motion stems from synchronizing when they decide to switch between the two modes. We propose a model for the individual decision making process and parametrize it from data. Next, we translate this individual-based model into its density-flow equations counterpart, considering 1D-motion along the group trajectory. Numerical solving these equations display a solitary wave propagating at constant speed. Coupling individual and collective levels, groups motion can then be seen as a wave propagating at some fraction of the individual walking speed even though each individual is at any moment either stopped or walking. Considering the minimal model embedded in these equations, we show analytically that it has the Korteweg-De Vries (KdV) Soliton as a steady regime solution. This soliton emerges from the non linear coupling of start/stop individual decisions which compensate exactly for diffusion and promotes a steady ratio of walking / stopped individuals, which in turn determines the wave speed. The convergence to only one solitary wave from any initial condition, and which can recover from perturbation, gives a high robustness to this biological system.

Update (12/18/17): I forgot to make a snarky remark along the following lines: One first needs to do a continuum approximation to get a relevant nonlinear wave equation.

For the actual sheep, one would think that there is some energy shedding as the wave propagates. :)

Tales from the ArXiv: It's Party Time!

You know you need sleep when... You read "parity-time" in the paper title "Gap solitons and symmetry breaking in parity-time symmetric microring CROWs" as "party-time".

Let's have a ball!

RIP Gerald Whitham (1927–2014)

Well, another one of the old guard from Caltech applied mathematics is gone. I found out earlier today that Gerald Whitham died on January 26th. Gerald was one of the pre-eminent early scholars in the study of nonlinear waves, which are near and dear to my heart.

I took Gerald's class AMa 98 when I was at Caltech --- that iteration of AMa 98 was the last class that he ever taught --- and that was the class that introduced me to solitons (so Whitham is the guy who showed me my first soliton). As many of you know, I have had a lot of fun in my career thinking about solitary waves (and, to a lesser extent, solitons). You can find one of those by-hand calculations in my scholarpedia entry on solitons (and related phenomena).

I was Gerald Whitham's last undergraduate advisee. This was in the sense of signing my cards so that I could take my classes --- we never worked on research together, though obviously some of his research interests rubbed off on me. He retired after my sophomore year (or at least rarely ever showed up to campus after that ... this page claims that he retired in 1998, but I think it was technically 1996), and my advisor was then switched over from Whitham to Oscar Bruno. Whenever I saw Whitham to get my card signed, he would inevitably complain about how the math department kept switching back and forth for the organization of the Math 107, 108, and 109 trifecta. (These switches occurred every decade or so, as far as I can tell. When it existed, Math 107 was a general introduction to analysis and topology, and then 108ab was analysis and 109ab was geometry. In the other form, 108abc was analysis and 109abc was geometry, where I believe that Math 107 was more or less the same as Math 108a.)

Update (2/22/14): Pasadena Star-News had an obituary for Whitham about a week ago.

Unearthed: A Report I Wrote on the KdV Equation During College

Oh, wow. From some 'random' googling, I just found the rough draft of a report on the KdV equation that I wrote my senior year in college. This apparently got uploaded when various past things from courses that Jerry Marsden taught were collected, scanned, and uploaded.

My exposition (including of the KdV equation!), my citation practices and being very thorough about giving proper credit for results and the exposition of others (right now, I think that my old report should have included specific statements of whose discussions I was following and not just citations to those papers), and my work on nonlinear waves has advanced a lot since that report -- though I think that one can see many hints of my current style in it. Certainly one can see some of my current research interests. I had completely forgotten about this old report. (My handwriting in parts of the report is almost exactly the same as now, though! :P)

If you want to compare my old exposition to my current exposition, see the Scholarpedia entry on solitons that I wrote with Norm Zabusky.

I should perhaps use this as before/after to show my students for certain assignments. I think this could be helpful for some interesting teaching opportunities.

I'm not sure why the rough draft was uploaded, though. Maybe Jerry's copy of the final draft got lost?

Scholarpedia Entry: "Soliton"

My scholarpedia entry on solitons is now officially live.

Hopefully, this will become the first port of call for information about solitary waves and solitons---that is certainly the intent of the article.

SklogWiki

Via some googling (because one of my articles is cited on one of the pages), I accidently found SklogWiki, a wiki dedicated to thermodynamics and statistical mechanics.

At the moment, it doesn't seem to be in particularly great shape. Scholarpedia seems to be in much better shape. (I really need to revise my soliton article and let the referees know that I have addressed their comments. That will have to occur after I return from Shanghai...)

Fermi, Pasta, Ulam, and the Birth of Experimental Mathematics

The May-June 2009 issue of the magazine American Scientist will include the article, Fermi, Pasta, Ulam and the Birth of Experimental Mathematics, which I coauthored with Norm Zabusky, Bambi Hu, and David Campbell. (I added an extra comma in the title of the blog entry, as I prefer that style. I was not allowed to use it in the article because of the magazine's official style policy.

This article is basically "FPU for Dummies", as it attempts to discuss the Fermi-Pasta-Ulam problem (and the massive amount of exciting work it has spawned; this includes multiple subdisciplines of math and physics!) without using mathematical equations. This is really hard for a subject like this...

Quote of the Day (another one)

After an unintentional bit of "awesomeness" in a paper I refereed today, I think I might have to replace the previous quote of the day. This whole situation started because of an unfortunate substitute for the word "data". Here is the quote:

"For many cases a localized date breaks up into a finite collection of solitary waves."

Indeed! I couldn't have said it better myself!

RIP Martin Kruskal (1925-2006)

I just got an e-mail from Norm Zabusky that Martin Kruskal died on December 26th, 2006. A brief obituary can be found here.

Kruskal did a lot of very important mathematical work, but I know him best for coauthoring an old PRL (from back in the day when PRLs were actually both seminal and understandable) with Norm that gave the first mathematical formulation of a soliton. Considering the fact that a huge component of my research involves solitons and solitary waves, we're talking about somebody who did some serious foundational stuff the stuff I study.

I unfortunately never had the chance to meet him. However, let me tell a brief story from the 2006 AMS/MAA Joint Meetings. Thankfully, Martin was around to receive his Steele Prize. I loved the fact that (1) he was the only person on the award podium who wore a t-shirt rather than something fancy and (2) he very humorously commented that the award came 30 years later than it should have. This was one of my major highlights of the meeting.

In the same e-mail, I was informed that another major person who laid the foundations for this field has lung cancer and isn't doing very well. (I'm going to suppress the name in public locations because I don't know how common this knowledge is.)

Because academia is such a small world, a certain component of it (namely, people in the fields I work---and, in some ways, especially people who laid the foundations of the field and whose work everybody knows) is one of my families in many respects. So, while this is not the same as somebody I know personal dying, there is still a strong connection here in other respects.