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. :)

"Granular Crystals: Nonlinear Dynamics Meets Materials Engineering"

Chiara Dario, Panos Kevrekidis, and I have written an expository article about granular crystals for the magazine Physics Today.

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.

"Solitary Matter Waves in Combined Linear and Nonlinear Potentials: Detection, Stability, and Dynamics"

One of my papers came out in its final published form today. Here are the details.

Title: Solitary matter waves in combined linear and nonlinear potentials: Detection, stability, and dynamics

Authors: Scott Holmes, Mason A. Porter, Peter Krüger, and Panayotis G. Kevrekidis

Abstract: We study statically homogeneous Bose-Einstein condensates with spatially inhomogeneous interactions and outline an experimental realization of compensating linear and nonlinear potentials that can yield constant-density solutions. We illustrate how the presence of a step in the nonlinearity coefficient can only be revealed dynamically and examine how to reveal it by exploiting the inhomogeneity of the sound speed with a defect-dragging experiment. We conduct computational experiments and observe the spontaneous emergence of dark solitary waves. We use effective-potential theory to perform a detailed analytical investigation of the existence and stability of solitary waves in this setting, and we corroborate these results computationally using a Bogoliubov–de Gennes linear stability analysis. We find that dark solitary waves are unstable for all step widths, whereas bright solitary waves can become stable through a symmetry-breaking bifurcation as one varies the step width. Using phase-plane analysis, we illustrate the scenarios that permit this bifurcation and explore the dynamical outcomes of the interaction between the solitary wave and the step.


As an additional note, there have been a couple of hundred theoretical/computational papers on BECs with spatially inhomogeneous nonlinearities, but (to my knowledge) there has been only a single experimental paper on the topic, and that paper's basic point was essentially just that one can actually make these things in the laboratory. The challenge is thus to do something interesting in the laboratory, and this paper includes some experimental designs to try to do that (and, indeed, it includes an experimentalist as one of the authors). So here's hoping that we'll see some of these things or other phenomena soon in spatially inhomogeneous BECs studied in laboratories...

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.

"Discrete Breathers in One-Dimensional Diatomic Granular Crystals"

A new paper by my collaborators and me has just appeared in Physical Review Letters. This is my 4th paper in PRL. The concerns discrete breathers in granular crystal, and it includes theory, numerical simulations, and (especially!) experiments. In fact, this paper gives the first experimental demonstration of intrinsic localized modes (these are the discrete breathers) in granular crystals, which is a rather exciting result. I believe that Caltech is going to be issuing a press release, so I'll pass along that and any ensuing press coverage later.

Title: Discrete Breathers in One-Dimensional Diatomic Granular Crystals

Authors: N. Boechler, G. Theocharis, S. Job, P. G. Kevrekidis, Mason A. Porter, and C. Daraio

Abstract: We report the experimental observation of modulational instability and discrete breathers in a one-dimensional diatomic granular crystal composed of compressed elastic beads that interact via Hertzian contact. We first characterize their effective linear spectrum both theoretically and experimentally. We then illustrate theoretically and numerically the modulational instability of the lower edge of the optical band. This leads to the dynamical formation of long-lived breather structures, whose families of solutions we compute throughout the linear spectral gap. Finally, we experimentally observe the manifestation of the modulational instability and the resulting generation of localized breathing modes with quantitative characteristics that agree with our numerical results.

Experimental Results Related to DNLS Equations

The final version of this paper is actually over a year old, but it was just published as a book chapter in a monograph written by one of my collaborators (Panos Kevrekidis). Panos wrote the first sections of the book, and then he invited a number of people to contribute individual chapters on more specific topics. He asked me to write a paper based on experiments relevant to discrete nonlinear Schrodinger (DNLS) equations because of the fact that I work closely with experimentalists on a number of topics. Hence, this paper is a review article that covers a theorist's view on experimental results. (Note that I purposely ran the paper by several experimental colleagues in relevant fields to ensure that I didn't say anything stupid.)

The title of the published version of the chapter (which you probably won't be able to download for free, which is why I included the link to the version on my website) has "DNLS" because the acronym has already been well-established by that point in the context of the book.

Keep your eyes on this spot for a number of additional papers. I have a bunch of stuff that's about to come out. Also, I have a comment to make related to the first-mover scientific advantage, but I'll leave that for a different blog entry because it relates to my networks research rather than nonlinear waves research.

The Porter of Seville

AKA: Lo que pasa en Sevilla se queda en Sevilla. (I'm pretty sure that one should use quedarse here rather than quedar. An online Spanish-English dictionary seems to back me up. Let me know if I got this horribly wrong.)

AKA: What happens in Seville stays in Seville.

Also, while my upcoming status as the Porter of Seville isn't as cool as The Rabbit of Seville or even The Barber of Seville, I think it's pretty damned good. Naturally, I earmarked that title for this particular blog entry when I first decided to attend this conference. (I don't normally come up with entry titles several months in advance, but this one was just too good to pass up.)

Early tomorrow morning, I will get on the bus to head to Heathrow airport for a morning flight to Madrid. I will then stay in the Madrid airport for a several-hour layover before I take a "one hour" flight to Seville. (I seriously should have found a better configuration.) I suspect that only about 30 minutes of that flight will have actual flying. I will be attending LENCOS, a conference on "Localized Excitations in Nonlinear Complex Systems". In practice, this will entail lots and lots of talks on solitary waves and allied concepts. This is rather narrow in scope mathematically, though a wide range of applications will hopefully (and likely) be represented. One thing I really like about the conference is that there are no parallel sessions and that it is a level playing field---every talk is 20 minutes + 5 minutes for questions. My past experience with relatively small conferences in which everybody sees every talk have been quite good. This format is usually quite conducive to meeting and have lots of chances to talk to other people, which can be particularly beneficial for young scientists who might otherwise be virtually ignored by their senior colleagues. There are supposed to be some posters as well (for people who chose to do things that way), but I can't see any reference to them on the website.

Interestingly, the conference venue was supposedly the inspiration for Carmen back in the day. I need to find a way to get "The Toreador Song" (or maybe "The March of the Toreadors", which would probably work better) into my talk. I will be presenting a talk on granular crystals.

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!

Matter-wave solitons with a periodic, piecewise-constant scattering length

Physical Review A just published one of my articles: Matter-wave solitons with a periodic, piecewise-constant scattering length.

My coauthors are A. S. Rodrigues, P. G. Kevrekidis, D. J. Frantzeskakis [who I met in real life for the first time at the conference I'm currently attending] , P. Schmelcher, and A. R. Bishop. [I have yet to meet Augusto Rodrigues, Peter Smelcher, and Alan Bishop in person.]

Here is our abstract: Motivated by recent proposals of “collisionally inhomogeneous” Bose-Einstein condensates (BECs), which have a spatially modulated scattering length, we study the existence and stability properties of bright and dark matter-wave solitons of a BEC characterized by a periodic, piecewise-constant scattering length. We use a "stitching" approach to analytically approximate the pertinent solutions of the underlying nonlinear Schrödinger equation by matching the wave function and its derivatives at the interfaces of the nonlinearity coefficient. To accurately quantify the stability of bright and dark solitons, we adapt general tools from the theory of perturbed Hamiltonian systems. We show that stationary solitons must be centered in one of the constant regions of the piecewise-constant nonlinearity. We find both stable and unstable configurations for bright solitons and show that all dark solitons are unstable, with different instability mechanisms that depend on the soliton location. We corroborate our analytical results with numerical computations.


The idea behind this is that my collaborators and I (as well as others) have done some work on BECs in "nonlinear lattices" in which the nonlinearity coefficient (which is proportional to the two-body scattering length) is a periodic function of space: g = g(x). In my past paper on this topic, my collaborators and I looked at periodic waves and g(x) given by a trig function. To try to delve more deeper into some things analytically, we decided to take a step back and let g(x) be a periodic step function. One can solve the governing partial differential equation in closed form in the g(x) = constant regions and then one can try to match the solutions at the boundaries between those regions. This was motivated by some conversations with more theoretical mathematicians (Bjorn Sandstede and Percy Deift) and to try to find a setup that would be more tractable to some theorem-proof work by people closer to the pure side of the mathematical spectrum. In the just-published paper, we looked at localized solutions. Our plan is to look at periodic solutions (in the form of elliptic functions) as well.

What happens in Rome stays in Rome...

...which of course has the obvious subtitle of "When in Rome, do like the Italians." :) (Ave Maria. Gee, it's good to see ya.)

I am going to the SIAM nonlinear waves conference, which is held once every two years. This is the third conference in the series. The first was in Orlando and the second was in Seattle.

Two of the conference organizers e-mailed me early on to ask my collaborators and me to organize a minisymposium. (As you can probably guess, this is one of the conferences where I know a ton of people.) We chose to organize a two-part session on nonlinear waves in periodic media, which is a topic that is near and dear to my heart.

I will let you know if I see anybody doing the Vatican Rag. (By the way, my musical jokes for the blog entry I write for my Vienna conference in September will be better. Some of you will see it coming from miles away, but I'll be doing it anyway.)

Highly Nonlinear Solitary Waves in Phononic Crystal Dimers

My paper, Highly Nonlinear Solitary Waves in Phononic Crystal Dimers, was just published as a "rapid communication" in Physical Review E. (For those of you who don't know the lingo, that means we almost pushed it into PRL but couldn't quite get it in there.)

My coauthors on the paper are theorist Panos Kevrekidis and experimentalists Chiara Daraio, Eric Herbold, and Ivan Szelengowicz (Chiara's grad student).

Here is the project in English: We have a chain of beads in which soft particles alternate with harder ones. We hit the thing and look at the properties of the pulse that goes through the system---basically, how its width and propagation speed depend on the properties of the beads and on having two different types of them (and the ensuing periodic structure) in the first place. We do analytics, numerics, and experiments and have some nice agreement between the three. (In fact, the analytics even got pretty technical in this paper.)

Here is the official abstract, which many of you will probably find much less understandable than what I just wrote (if you look closely, however, you can see that it basically says the same thing as what I just wrote---just without the technical terms that make things more precise): We investigate the propagation of highly nonlinear solitary waves in heterogeneous, periodic granular media using experiments, numerical simulations, and theoretical analysis. We examine periodic arrangements of particles in experiments in which stiffer and heavier beads (
stainless steel) are alternated with softer and lighter ones
(polytetrafluoroethylene beads). We find good agreement between experiments and numerics in a model with Hertzian interactions between adjacent beads, which in turn agrees very well with a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments and general bead interactions. Our analysis encompasses previously studied examples as special cases and also provides key insights into the influence of the dimer lattice on the properties
width and propagation speed of the highly nonlinear wave solutions.