What Happens in Boston Stays in Boston

I'm on my way to Boston for the workshop to celebrate David Campbell's 80th birthday! It will surely be chaotic. :)

"Recurrence Recovery in Heterogeneous Fermi–Pasta–Ulam–Tsingou Systems"

Another of my papers was published in final form today. Here are some details.

Title: Recurrence Recovery in Heterogeneous Fermi–Pasta–Ulam–Tsingou Systems

Authors: Zidu Li, Mason A. Porter, and Bhaskar Choubey

Abstract: The computational investigation of Fermi, Pasta, Ulam, and Tsingou (FPUT) of arrays of nonlinearly coupled oscillators has led to a wealth of studies in nonlinear dynamics. Most studies of oscillator arrays have considered homogeneous oscillators, even though there are inherent heterogeneities between individual oscillators in real-world arrays. Well-known FPUT phenomena, such as energy recurrence, can break down in such heterogeneous systems. In this paper, we present an approach—the use of structured heterogeneities—to recover recurrence in FPUT systems in the presence of oscillator heterogeneities. We examine oscillator variabilities in FPUT systems with cubic nonlinearities, and we demonstrate that centrosymmetry in oscillator arrays may be an important source of recurrence.

What Happens at "Snowbird" Stays at "Snowbird"

Today I am off to the "Snowbird Meeting" (aka the SIAM applied-dynamical systems conference) for the latest instantiation of my favorite scientific conference series.

"Nanoptera in Higher-Order Nonlinear Schrödinger Equations: Effects of Discretization"

A paper of mine has just been published in final form. Here are some details about it.

Title: Nanoptera in Higher-Order Nonlinear Schrödinger Equations: Effects of Discretization

Authors: Aaron J. Moston-Duggan, Mason A. Porter, and Christopher J. Lustri

Abstract: We consider generalizations of nonlinear Schrödinger equations, which we call “Karpman equations,” that include additional linear higher-order derivatives. Singularly- perturbed Karpman equations produce generalized solitary waves (GSWs) in the form of solitary waves with exponentially small oscillatory tails. Nanoptera are a special type of GSW in which the oscillatory tails do not decay. Previous research on continuous third-order and fourth-order Karpman equations has shown that nanoptera occur in specific settings. We use exponential asymptotic techniques to identify traveling nanoptera in singularly-perturbed continuous Karpman equations. We then study the effect of discretization on nanoptera by applying a finite-difference discretization to continu- ous Karpman equations and examining traveling-wave solutions. The finite-difference discretization turns a continuous Karpman equation into an advance–delay equation, which we study using exponential asymptotic analysis. By comparing nanoptera in these discrete Karpman equations with nanoptera in their continuous counterparts, we show that the oscillation amplitudes and periods in the nanoptera tails differ in the continuous and discrete equations. We also show that the parameter values at which there is a bifurcation between nanopteron solutions and decaying oscillatory solutions depends on the choice of discretization. Finally, by comparing different higher-order discretizations of the fourth-order Karpman equation, we show that the bifurcation value tends to a nonzero constant for large orders, rather than to 0 as in the associated continuous Karpman equation.

I Have "NoIDEA"

It turns out that there is a journal called "Nonlinear Differential Equations and Applications". The acronym it uses is "NoDEA".

I feel inspired!

I am going to start a journal called "Nonlinear Integro-Differential Equations and Applications", and its acronym will be "NoIDEA"!

— Mason Porter (@masonporter) September 18, 2019

"Nonlinear Excitations in Magnetic Lattices with Long-Range Interactions"

A paper of mine appeared in final form a few days ago. Here are the details.

Title: Nonlinear Excitations in Magnetic Lattices with Long-Range Interactions

Authors: Miguel Molerón, Chris Chong, Alejandro J. Martínez, Mason A. Porter, Panayotis G. Kevrekidis, and
Chiara Daraio

Abstract: We study—experimentally, theoretically, and numerically—nonlinear excitations in lattices of magnets with long-range interactions. We examine breather solutions, which are spatially localized and periodic in time, in a chain with algebraically-decaying interactions. It was established two decades ago (Flach 1998 Phys. Rev. E 58 R4116) that lattices with long-range interactions can have breather solutions in which the spatial decay of the tails has a crossover from exponential to algebraic decay. In this article, we revisit this problem in the setting of a chain of repelling magnets with a mass defect and verify, both numerically and experimentally, the existence of breathers with such a crossover.

Tales from the ArXiv: Grandma Got Run Over By a Reindeer

If I had done this study, I would make a video submission to the Gallery of Nonlinear Images (may it rest in peace), and I would set it to the tune of the song Grandma Got Run Over By a Reindeer. Just saying...

812 Double Pendulums with Different Initial Conditions

This is what 812 double pendulums with different initial conditions look like https://t.co/jklEvfb5EI | https://t.co/ctsPCmFfEJ pic.twitter.com/jMar6rlRma

— Massimo (@Rainmaker1973) October 26, 2018

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.

"Nanoptera in a Period-2 Toda Chain"

One of my papers recently came out in final form (with its volume and all of its other publication coordinates). Here are some details.

Title: Nanoptera in a Period-2 Toda Chain

Authors: Christopher J. Lustri and Mason A. Porter

Abstract: We study asymptotic solutions to a singularly perturbed, period-2 Toda lattice and use exponential asymptotics to examine "nanoptera," which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains. With this approach, we isolate the exponentially small, constant-amplitude waves, and we elucidate the dynamics of these waves in terms of the Stokes phenomenon. We fi nd a simple asymptotic expression for these waves, and we study con figurations in which these waves vanish, producing localized solitary-wave solutions. In the limit of small mass ratio between the two types of particles in the lattice, we derive a simple antiresonance condition for the manifestation of such solutions.

"Hyperchaos" in the White House

I saw "CHAOS" trending on Twitter this morning, so I automatically looked it up, before seeing that it was about politics and then moving on to something else.

And now I see this article, which predominantly consists of an interview with nonlinear dynamicist (and pioneer of chaos) Jim Yorke, including discussions about both mathematical chaos and hyperchaos.

Here is how the part with Yorke begins: "That's not chaos, according to James A. Yorke, Distinguished University Professor of Mathematics and Physics at the University of Maryland at College Park." It goes on from there.

Just think about it: Jim Yorke, an expert in chaotic dynamics, was interviewed by CNN about Donald Trump, precisely because of the former's expertise in mathematical chaos. Yup, we've gone full Illuminati.

(Tip of the cap to Bruno Eckhardt and fuzzy sweatshirt particle.)

Mathematical Haiku

My mathematical haiku are part of a collection of haiku published in Journal of Humanistic Mathematics in January 2018: https://t.co/ZbG0JEBgL6

(Yes, even my haiku sometimes include sarcasm.) pic.twitter.com/ogPXYmmR9W

— Mason Porter (@masonporter) February 22, 2018

(Thanks to Paul Glendinning for the Twitter 'mention', from which I learned that my haiku made it into the article.)
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"Direct Measurement of Superdiffusive Energy Transport in Disordered Granular Chains"

Our paper in which we — after many years of effort — showed strongly nonlinear Anderson phenomena in experiments is finally out in published form!

Here are some details.

Title: Direct Measurement of Superdiffusive Energy Transport in Disordered Granular Chains

Authors: Eunho Kim, Alejandro J. Martínez, Sean E. Phenisee, Panayotis G. Kevrekidis, Mason A. Porter, and Jinkyu Yang

Teaser: Wave propagation is often nonlinear in character, yet the interplay between disorder and nonlinearity remains elusive. Kim et al. use experiments and corroborating numerical simulations to investigate this phenomenon and demonstrate superdiffusive energy transport in disordered granular chains.

Abstract: Energy transport properties in heterogeneous materials have attracted scientific interest for more than half of a century, and they continue to offer fundamental and rich questions. One of the outstanding challenges is to extend Anderson theory for uncorrelated and fully disordered lattices in condensed-matter systems to physical settings in which additional effects compete with disorder. Here we present the first systematic experimental study of energy transport and localization properties in simultaneously disordered and nonlinear granular crystals. In line with prior theoretical studies, we observe in our experiments that disorder and nonlinearity—which individually favor energy localization—can effectively cancel each other out, resulting in the destruction of wave localization. We also show that the combined effect of disorder and nonlinearity can enable manipulation of energy transport speed in granular crystals. Specifically, we experimentally demonstrate superdiffusive transport. Furthermore, our numerical computations suggest that subdiffusive transport should be attainable by controlling the strength of the system’s external precompression force.

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.

What's a Rouge Wave?

I managed to sneak one of the all-time best lines from Buffy/Angel into the tweet below.

At first, I accidentally, introduced the typo "rouge", but I managed to change the tweet before anybody could reply with "What's a rouge wave?" Naturally, this inspired the title of this post.

Rogue Wave Hunters: Wind-generated waves in ring-shaped water tanks can spontaneously yield behemoth waves: https://t.co/4QSaR0MsSl

— DynamicalSystemsSIAM (@DynamicsSIAM) April 7, 2017

Advertisement: Faculty Position in Networks and Nonlinear Systems, Mathematical Institute, University of Oxford

Oxford has now advertised my old job, which is for an Associate Professorship or Professorship in the Mathematical Institute. The advertisement is for somebody in Networks and Nonlinear Systems.

Go apply for it!

Tales from the ArXiv: "With Inertia and Frustration"

The title of a new paper on the arXiv this morning is "Bifurcations and singularities for coupled oscillators with inertia and frustration".

I could add the phrase "with inertia and frustration" to the end of the title of just about any of my papers, and it would be accurate.

(OK, both terms have specific technical meaning in this case, but work with me.)

What Happens in Columbus Stays in Columbus (2016 Edition)

I had heading to Columbus, OH to be in residence at the Mathematical Biosciences Institute (MBI) at The Ohio State University for the next month as part of a semester-long emphasis program on dynamics of biologically-inspired networks that I helped organize. I also co-organized workshop 3 (on generalized network structures and dynamics) and will participate in workshops 3 and 4 (on control and observability of network dynamics).

You can request a live stream of these workshops!

March Meeting Chatchkes

I was pretty selective about chatchkes, but here is the stuff I have gotten so far –– most of them are appropriately GSNP-themed. GSNP is the Group on Statistical and Nonlinear Physics, which is my main home in this community. (I did get roped into signing up for the Soft Matter group, and then I was given appropriately granular mints.)

The rattleback was my reward for the Plinko game. It was what I wanted; there were multiple possible chatchkes that one could get, but the game was less random than a good Plinko construction would have been.

The coolest chatchke by far is the hot-pack/cold-pack, which is also appropriately granular (and which I think can be appropriated for many other uses in addition to the intended ones).

By the way: I plan to use everything in this picture, with the possible exception of the brain.

At registration, everybody received a keychain with a small bottle of antibacterial hand sanitizer (because of the scientific reputation for hygiene).

What Happens at the March Meeting Stays at the March Meeting (2016 Edition)

I am heading to Baltimore to go to the 2016 APS March Meeting along with 10000 or so of my best friends. I haven't been to this conference since 2010.

Tomorrow I'll be giving an invited talk on localized modes in granular crystals, and on Wednesday I'll be chairing one of the networks sessions.

Update (3/17/16): You can now download the slides of my talk, which is about one-dimensional granular crystals.