"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...

"Dark Solitary Waves in a Class of Collisionally Inhomogeneous Bose-Einstein Condensates"

A new paper of mine just got published in final form today. It's my first paper on Bose-Einstein condensates since 2008 (or, as Georg Gottwald called it, "a relapse"). Anyway, here are the details.


Title: Dark Solitary Waves in a Class of Collisionally Inhomogeneous Bose-Einstein Condensates

Authors: Chang Wang, Kody J. H. Law, Panayotis G. Kevrekidis, and Mason A. Porter

Abstract: We study the structure, stability, and dynamics of dark solitary waves in parabolically trapped, collisionally inhomogeneous Bose-Einstein condensates (BECs) with spatially periodic variations of the scattering length. This collisional inhomogeneity yields a nonlinear lattice, which we tune from a small-amplitude, approximately sinusoidal structure to a periodic sequence of densely spaced spikes. We start by investigating time-independent inhomogeneities, and we subsequently examine the dynamical response when one starts with a collisionally homogeneous BEC and then switches on an inhomogeneity either adiabatically or nonadiabatically. Using Bogoliubov-de Gennes linearization as well as direct numerical simulations of the Gross-Pitaevskii equation, we observe dark solitary waves, which can become unstable through oscillatory or exponential instabilities. We find a critical wavelength of the nonlinear lattice that is comparable to the healing length. Near this value, the fundamental eigenmode responsible for the stability of the dark solitary wave changes its direction of movement as a function of the strength of the nonlinearity. When it increases, it collides with other eigenmodes, leading to oscillatory instabilities; when it decreases, it collides with the origin and becomes imaginary, illustrating that the instability mechanism is fundamentally different in wide-well versus narrow-well lattices. When starting from a collisionally homogeneous setup and switching on inhomogeneities, we find that dark solitary waves are preserved generically for aligned lattices. We briefly examine the time scales for the onset of solitary-wave oscillations in this scenario.

Tales from the ArXiv: Daft Punk and Ultracold Atoms

The title of this article has a nice allusion to Daft Punk. The title is Production of quantum degenerate strontium gases: Larger, better, faster, colder. Nice!

"The Condensed Matter Song"

"The Condensed Matter Song"

By Mason A. Porter, Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford

(parody of the "Matter-Patter Trio" by Gilbert and Sullivan)



I'm having lots of trouble with this awful calculation –
So I shall go back to my lab for some Bose-Einstein condensation.
And see if better vortex lattices cannot attract my senses,
And with some extra stirring I shall find the consequences.
Oh but in my complex conjugate I did forget the dagger,
And now that I get vortices I have regained my swagger,
And a word or two of compliment my vanity would flatter,
But I won't get tenure anyway, because it's really condensed matter!

Because atomic physics is really condensed matter,
condensed matter, condensed matter,
Yes it's really condensed matter,
condensed matter, condensed matter,

Yes it's really condensed matter,
It is really condensed matter,

BECs are condensed matter, condensed matter, condensed matter!

matter, condensed matter, condensed matter,
condensed matter, condensed matter, ...


It may be a little tired and perhaps generally silly
And that energy landscape seems to me complex and hilly;
Though with it I want to grapple with a biological question,
But I know not what, so do you have a suggestion?
I shall then submit a paper to Physics Review Letters,
With excellent predictions to be tested when technology is better,
But right now I wish that landscape would be flatter,
And I don't actually know biology, so I study condensed matter!

Yes I study condensed matter,
condensed matter, condensed matter,
Biology is really condensed matter,
condensed matter, condensed matter,

Yes it's really condensed matter,
It is really condensed matter,

Biology is condensed matter, condensed matter, condensed matter!

matter, condensed matter, condensed matter,
condensed matter, condensed matter, ...


If I had been so lucky I would have studied complex systems
But unless I find a power law surely nobody will listen –
I was given good advice when my mentor saw me erring
That to study networks and call it "physics" would be ever so daring,
And then with real data I would truly get to fiddle,
And publish papers in Nature and Science that aren't worth a piddle.
That particularly vapid, unintelligible patter
Is very sexy nowadays, but we all know it's really condensed matter!

Yes it's really condensed matter,
condensed matter, condensed matter,
Complex systems is really condensed matter,
condensed matter, condensed matter,


All this rapid-publication, unintelligible patter
Is presented at the March Meeting, so it must be condensed matter,
Yes this rapid-publication, unintelligible patter
Is all part of the subject that we call condensed matter,
matter, condensed matter, condensed matter,
matter, condensed matter, condensed matter!



(OK, so I'm probably being a little harsh here and I'm definitely biting the hand that feeds me. But I couldn't resist doing this...)

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.

Tales from the arXiv: the SLAP technique

The following abstract makes me giggle:


arXiv:0811.3409
Date: Thu, 20 Nov 2008 20:39:04 GMT (1009kb)

Title: Coherent Patterning of Matter Waves with Subwavelength Localization
Authors: J. Mompart, V. Ahufinger, G. Birkl
Categories: quant-ph
Comments: 5 pages, 5 figures
\\
We introduce the Subwavelength Localization via Adiabatic Passage (SLAP)
technique to coherently achieve state-selective patterning and addressing of
matter waves. The SLAP technique consists in coupling two partially overlapping
and spatially structured laser fields to three internal levels of the matter
wave yielding state-selective localization at those positions where the
adiabatic passage process does not occur. We show that by means of this
technique matter wave localization down to the single nanometer can be
achieved. We analyze in detail the potential implementation of the SLAP
technique for nano-lithography with an atomic beam of metastable Ne* and for
coherent patterning of a two-component 87Rb Bose-Einstein condensate.
\\ ( http://arxiv.org/abs/0811.3409 , 1009kb)

Vienna Calling (aka: What happens in Vienna stays in Vienna)

I seem to be doing just about everything possible (including finishing unpacking from my recent apartment change!) to avoid finishing the .ppt slides for my talk on Wednesday. I really ought to be much better about this, because I leave tomorrow morning for Vienna and if I don't finish the talk today, I'll need to eat into some of my exploration time tomorrow. (Granted, I'll have plenty more exploration time on other days...)

I am attending a workshop at the Wolfgang Pauli Institute called The Gross-Pitaevskii equation and its application for Bose-Einstein condensates in optical lattices. There are only 9 talks, though I think there were originally going to be around 25 of them. This could make the workshop either really good or really bad. Lots of discussion time has been allocated, and with the small number of people I suspect we'll even be able to calculate some stuff and get going on some projects. That would be really great! However, I can also envision things backfiring by everybody just going their own way and not interacting enough. Let's see how it goes. Even in the worst case, it would also just mean more time to explore Vienna, which isn't such a bad thing either (to say the least).

Anyway, my hope is to have a very exciting trip both academically and culturally. I am particularly excited about exploring the city's fine musical tradition! If I get a chance, I might even try to make my way here to pay my respects.

This will be my first trip to Austria. As for when I get back to Oxford, on Sunday I'll be back (so to speak).

Averaging of Nonlinearity Management with Dissipation

My paper called Averaging of nonlinearity management with dissipation just appeared in Physical Review A. My coauthors are S. Beheshti, K. J. H. Law, and P. G. Kevrekidis (all from U. Mass. Amherst).

This paper is a short but interesting bit of research whose motivation is best described with its abstract:

Motivated by recent experiments in optics and atomic physics, we derive an averaged nonlinear partial differential equation describing the dynamics of the complex field in a nonlinear Schrödinger model in the presence of a periodic nonlinearity and a periodically varying dissipation coefficient. The incorporation of dissipation in our model is motivated by experimental considerations. We test the numerical behavior of the derived averaged equation by comparing it to the original nonautonomous model in a prototypical case scenario and observe good agreement between the two.


Basically, the point is that the theoretical research involving "nonlinearity management" (which typically means using temporal and/or spatial adjustment on one or more of the nonlinear terms in a partial differential equation) uses an entirely conservative setting. However, one one applies one of these strategies experimentally, there is typically an additional dissipation mechanism that gets introduced. (We saw this in our work with experimentalists on nonlinearity management in the setting of nonlinear optics. This short paper was especially motivated by the associated series of experiments, though it's relevant for some Bose-Einstein condensate stuff as well.) Our paper take some theory for the conservative case and works it out for the periodic dissipation case that we saw in our optics experiments as an example of how one would do this more generally.

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.

Vortex Lattice Locking in Two-Component Bose-Einstein Condensates

This is the title of my new paper that just came out. You can find a link here.

Unlike most of my papers, this one is very condensed-mattery (to invent a new word... think of it the way you would use the word "buttery") and doesn't have much mathematics at all. I do spend a lot of time working on condensed matter physics problems, but the style of my papers ordinarily reveals my nonlinear heart (or, rather, my nonlinear science/applied math heart). This particular paper was written jointly with my fellow scientists in Caltech's theoretical condensed matter group, and the flavor of the paper is correspondingly different---it's most definitely a physics paper rather than an applied math paper.

The first author is Caltech postdoc Ryan Barnett. Also on the list are Caltech faculty member Gil Refael and non-Caltech condensed matter theorist Hanspeter buchler (who goes by Hans Peter in publications).

Here is the abstract:

The vortex density of a rotating superfluid, divided by its particle
mass, dictates the superfluid’s angular velocity through the Feynman relation. To
find how the Feynman relation applies to superfluid mixtures, we investigate a
rotating two-component Bose–Einstein condensate, composed of bosons with
different masses. We find that in the case of sufficiently strong interspecies
attraction, the vortex lattices of the two condensates lock and rotate at the drive
frequency, while the superfluids themselves rotate at two different velocities,
whose ratio equals the ratio between the particle masses of the two species.
In this paper, we characterize the vortex-locked state, establish its regime of
stability, and find that it survives within a disk smaller than a critical radius,
beyond which vortices become unbound and the two Bose-gas rings rotate
together at the frequency of the external drive.


I view this as a form of synchronization, but instead of the better-studied frequency-locking, one actually needs to take the different masses of the species into account (it's a momentum-locking). One of the very important points we make is that if you look at a lot of papers, you'll see that they assume that two different masses are the same "without loss of generality" (and the literature on multiple-component Bose-Einstein condensates is absolutely riddled with that assumption). Of course, as we show in this paper, that's just not true, as there are very interesting phenomena that you'll simply miss if you always make that assumption.

What happens in Durham stays in Durham

No, I am not referring to the one in North Carolina, so I won't be able to visit the people I know in that state.

I'll be visiting University of Durham tomorrow and Wednesday and will be giving a talk in their atomic and molecular physics seminar series. (I'll be speaking about some of my work on Bose-Einstein condensates.)

For what it's worth, however, one of the people taking me to dinner tomorrow evening is a current graduate student at Durham who was an undergrad at Duke (i.e., in the other Durham).

I haven't revised my slides since the last time I gave a BEC talk, but I'm hoping to find a way to get the word "lollygagging" in it somewhere. (Hmmmm... I guess none of the UK people reading this are going to recognize this allusion.)

Update: It turns out that the graduate student I mentioned above first encountered my name a few years ago because of a preprint I wrote back in the day giving an introduction to LaTeX for people using LaTeX for the first time. (This preprint, which you can find here, has actually gotten a fair bit of circulation over the years. I wonder if more people have read that paper than any of my other papers? I've been meaning to update that article for several years because I know a lot more about LaTeX than I did back then, but it's very far down on the list as far as my scientific endeavors are concerned. Maybe I'll try to find an interested student to do that at some point just because while the article is already very useful, it would be nice to make it even more useful.)

On the way to Durham, I switched lines in a station in Birmingham --- does anybody remember if the train line that goes through Birmingham, AL also hits Durham, NC? I passed through a station in York. It's sometimes easy to forget that there was an old York. :) (Even though I have a dessert that hails from Yorkshire when I eat prime rib at Lawry's...)

"Modulated Amplitude Waves in Collisionally Inhomogeneous Bose-Einstein Condensates"

One of my research papers just got published in Physica D. My collaborators and I posted a version of it on the arXiv many moons ago and, in fact, it has already been cited a couple of times.

My coauthors for this paper are Panos Kevrekidis, Boris Malomed, and Dimitri Frantzeskakis.

Here is the abstract:

We investigate the dynamics of an effectively one-dimensional Bose–Einstein condensate (BEC) with scattering length a subjected to a spatially periodic modulation, a = a (x ) = a (x + L ). This “collisionally inhomogeneous” BEC is described by a Gross–Pitaevskii (GP) equation whose nonlinearity coefficient is a periodic function of x . We transform this equation into a GP equation with a constant coefficient and an additional effective potential and study a class of extended wave solutions of the transformed equation. For weak underlying inhomogeneity, the effective potential takes a form resembling a superlattice, and the amplitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlinear generalization of the Ince equation. In the small-amplitude limit, we use averaging to construct analytical solutions for modulated amplitude waves (MAWs), whose stability we subsequently examine using both numerical simulations of the original GP equation and fixed-point computations with the MAWs as numerically exact solutions. We show that “on-site” solutions, whose maxima correspond to maxima of a (x ), are more robust and likely to be observed than their “off-site” counterparts.


Basically, the idea is that the nonlinearity coefficient is periodic (which can be achieved using a spatially-periodic magnetic field) and one can get lots of interesting things by putting the periodicity there instead of in the linear potential (which has been studied in considerable detail by many people, including me).

Quasiperiodic Dynamics in Bose-Einstein Condensates

Here's another technical post...

This paper is a project dating back to my Georgia Tech days. We first submitted this for publication in June 2004 and ended up needing to submit it to another journal. (The referee for the first journal never actually rejected us, but we ended up deciding that we wouldn't be able to please him and that we should instead just submit the paper somewhere else.)

My coauthors here are all mathematicians who write almost all of their papers in theorem-proof style and this is my only true theorem-proof paper to date. (I have a couple other papers that have some small proofs, but this entire paper is in the theorem-proof style.) They are Shui-Nee Chow, Yingfei Yi, and Martijn van Noort. (Martijn has since left science to pursue other things).

This project arose because one Shui-Nee Chow noticed that the ordinary differential equations for BEC standing waves that I was studying were very similar to the forced one degree-of-freedom Hamiltonian systems he, Martin, and Yingfei were studying theoretically. They needed an example and I could use some theorems, so this became a classic you-put-your-theorems-in-my-BECs (Bose-Einstein condensates) situations. We worked together on this for a semester, submitted the paper initially when Martijn left Georgia Tech for another postdoc, and the rest is history.

Here is the abstract:

We employ KAM theory to rigorously investigate quasiperiodic dynamics in cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and superlattices. Toward this end, we apply a coherent structure ansatz to the Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation describing the spatial dynamics of the condensate. For shallow-well, intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather sets to prove that one obtains mostly quasiperiodic dynamics for condensate wave functions of sufficiently large amplitude, where the minimal amplitude depends on the experimentally adjustable BEC parameters. We show that this threshold scales with the square root of the inverse of the two-body scattering length, whereas the rotation number of tori above this threshold is proportional to the amplitude. As
a consequence, one obtains the same dynamical picture for lattices of all depths, as an increase in depth essentially affects only scaling in phase space. Our approach is applicable to periodic superlattices with an arbitrary number of rationally dependent wave numbers.


The work I had done before this paper (the papers on period-multiplied solutions I had written with Predrag Cvitanovic') concentrated on situations with small amplitude periodic potentials, and this one instead considered a different near-integrable situation.

One interesting thing to do to get an idea of the range my work can cover on the math--physics spectrum is to compare the phrasing of the abstracts in this paper and the other one about which I blogged below. If you want an even better idea, compare how things are phrased in the two papers. (This actually leaves a big part out of the "physics" spectrum, as I also have papers which concentrate completely on real data and others which have both real data and experiments---though I've never been the one who actually does any of the experiments. That said, I am starting to have serious thoughts of eventually having an applied math lab, as my phononic crystals collaborator at Caltech is an experimentalist but had a theorist as a Ph.D. advisor and although that theorist isn't in a math department, he's basically an applied mathematician. I know others who have done this without any experimental training, so I may well do this at some point. For BECs, it won't be possible, but for tabletop things like the chains of beads it is definitely feasible.)

Fractional-period excitations in continuum periodic systems

This paper just appeared in Physical Review A. My coauthors and I originally sent a shorter version of this paper to PRL. In the words of one of them, it kicked off the goalpost. One of the referees indicated to publish it 'as is' and the other asked for us to address some concerns. We attempted to do this; he/she thanked us for the obviously large amount of effort it took to do this. (This referee did his/her job. We doubted a couple of points at first, but he/she was right on every single point. I have to give credit where credit is due.) In our opinion, we satisfied everything, but the referee wasn't convinced and suggested we add more details in appendices to turn it into a PRA article. A new referee also suggested PRA because, in his/her view, no paper without any experiments should ever be published in PRL. WTF? (This person needs to check out the journal's masthead and quite a few of the past articles that have been published in it.)

We could have brought that up with the editors, but it probably would have been a long fight anyway (and the chance of success didn't seem that great because this would have involved at least one more review) because of the other referee, so we decided it wasn't worth the trouble. So, continuing the soccer terminology, after the goal post, instead of taking another shot at the goal, we got nailed against a fence and have a PRA instead of a PRL. (The original plan was to have the PRL and then submit an archival paper to a SIAM journal, but we instead just have a physicsy version of the archival paper in PRA.) I'm happy for the PRA, but there's a bit of an issue of what might have been. (To provide context, by the way, this was moved over from PRL to PRA before either of my two PRLs got accepted, so at the time we were dealing with this, this had been my best shot ever at getting a PRL. I.e., it matters much less now than it seemed to at the time.)

OK, so now that I've resuscitated a dead rant (braaaaaaaains...) that is no longer relevant (if it ever was overly relevant in the first place), let's acknowledge my coauthors and discuss the science a bit.

My main coauthors on this paper were Hector Nistazakis, Panos Kevrekidis, and Dimtri Frantzeskakis, who are all members of the Greek BEC mafia. Alex Nicolin was a theoretical consultant and a familiar person (Jit Kee Chin '01) was an experimental consultant. Some experiments were performed in an afternoon or two in the Ketterle lab, but they didn't work out and the equipment had to be used for experiments in the group's official agenda, so the project ended up only having theoretical and computational components. (We were hoping to try to observe our stuff experimentally, but it didn't work. It can work in theory, and I hope to see that happen someday.)

Here is the abstract:

We investigate the generation of fractional-period states in continuum periodic systems. As an example, we consider a Bose-Einstein condensate confined in an optical-lattice potential. We show that when the potential is turned on nonadiabatically, the system explores a number of transient states whose periodicity is a fraction of that of the lattice. We illustrate the origin of fractional-period states analytically by treating them as resonant states of a parametrically forced Duffing oscillator and discuss their transient nature and potential observability.


You'll notice that Bose-Einstein condensates (BECs) aren't actually mentioned in the title of the paper. (Note: I've provided a brief explanation of BECs in a prior post, so I'll just let you google that if interested.) The reason is that this is a very general phenomenon that can occur in continuum systems (modeled by partial differential equations) which also have some sort of periodicity in the a dependent variable that turns a continuous translational symmetry into a discrete one. BECs, which can be modeled by a nonlinear Schrödinger (NLS) equation known as the Gross-Pitaevskii (GP) equation, provided our focus example.

In the presence of a periodic potential (which is an optical lattice for BECs), a natural thing to do is to look for solutions of the same period (using Bloch theory, for example). For BECs, one can construct such solutions using either the GP equation or with a discrete model (such as the Bose-Hubbard model) in which the lattice lengthscale is imposed on the model via the discretization. One can also construct solutions whose periodicity is a multiple of the lattice period, which I did analytically using Hamiltonian perturbation theory (and KAM considerations) in previous papers. Alex saw these things numerically using a discrete NLS equation in a paper that came out at the same time. I think my eventual-PRE and his eventual-PRA may even have been posted on the arxiv on consecutive days. (We saw each other's papers and started talking to each other.)

About a year after my work was published, the Chu group at Stanford constructed period-doubled states experimentally. (His group was motivated by Alex's paper, as that came out of the Pethick-Smith BEC group, and my collaborator and I were completely unknown in the community, as we come instead from the nonlinear science community.) I was really excited when that paper was posted on the arXiv, especially given that when I spoked about my work at the 2004 March Meeting, there seemed to be some skepticism in the audience as to whether such states could actually be observed. (One of my big messages for that paper is that although Bloch theory gives canonical solutions whose lengthscale matches the one imposed by the lattice that from a dynamical systems perspective, other types of states were also natural even though people trained in atomic physics might not expect it.)

Fractional-period states can be constructed similarly. (I used multiple-scale perturbation theory for the analytics.) If one ignores the harmonic trap, one can construct them. With the harmonic trap, they arise as potentially long-lived transient solutions. (We studied the situation with the trap numerically and that without the trap both analytically and numerically.) A discrete NLS cannot possibly pick them up a priori because the lattice lengthscale is imposed in the modeling. It's not always clear when the GP is a better description and when the Bose-Hubbard model (or another discrete NLS equation) is a better description for the macroscopic dynamics of BECs, so it's really nice to show a solution that one can have and the other can't. Obviously, observing this stuff experimentally will really be nice, but I don't think it will be easy and most of the BEC labs have moved on to things like fermions, so we'll see if this ever comes to pass.