"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.

"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.

"Nanoptera in Weakly Nonlinear Woodpile Chains and Diatomic Granular Chains"

A paper of mine was just published in final form. Here are some details.

Title: Nanoptera in Weakly Nonlinear Woodpile Chains and Diatomic Granular Chains

Authors: Guo Deng, Christopher J. Lustri, and Mason A. Porter

Abstract: We study ``nanoptera," which are nonlocalized solitary waves with exponentially small but nondecaying oscillations, in two singularly perturbed Hertzian chains with precompression. These two systems are woodpile chains (which we model as systems of Hertzian particles and springs) and diatomic Hertzian chains with alternating masses. We demonstrate that nanoptera arise from the Stokes phenomenon and appear as special curves (called Stokes curves) are crossed in the complex plane. We use techniques from exponential asymptotics to obtain approximations of the oscillation amplitudes. Our analysis demonstrates that traveling-wave solutions in a singularly perturbed woodpile chain have a single Stokes curve, which generates oscillations behind the wave front. Comparing these asymptotic predictions with numerical simulations reveals that our asymptotic approximation accurately describes the nondecaying oscillatory behavior in a woodpile chain. We perform a similar analysis of a diatomic Hertzian chain, and we show that each nanopteron solution has two distinct exponentially small oscillatory contributions. We demonstrate that there exists a set of mass ratios for which these two contributions cancel to produce localized solitary waves. This result builds on prior experimental and numerical observations that there exist mass ratios that support localized solitary waves in diatomic Hertzian chains without precompression. Comparing our asymptotic and numerical results for a diatomic Hertzian chain with precompression reveals that our exponential asymptotic approach accurately predicts the oscillation amplitude for a wide range of system parameters, but it fails to identify several values of the mass ratio that correspond to localized solitary-wave solutions.

"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.

"Variability in Fermi–Pasta–Ulam–Tsingou Arrays Can Prevent Recurrences"

A paper of mine came out in final form today. Here are some details.

Title: "Variability in Fermi–Pasta–Ulam–Tsingou Arrays Can Prevent Recurrences"

Authors: Heather Nelson, Mason A. Porter, and Bhaskar Choubey

Abstract: In 1955, Fermi, Pasta, Ulam, and Tsingou reported recurrence over time of energy between modes in a one-dimensional array of nonlinear oscillators. Subsequently, there have been myriad numerical experiments using homogenous FPUT arrays in the form of chains of ideal, nonlinearly coupled oscillators. However, inherent variations (e.g., due to manufacturing tolerance) introduce heterogeneity into the parameters of any physical system. We demonstrate that such tolerances degrade the observance of recurrences, often leading to complete loss in moderately-sized arrays. We numerically simulate heterogeneous FPUT systems to investigate the effects of tolerances on dynamics. Our results illustrate that tolerances in real nonlinear oscillator arrays may limit the applicability of results from numerical experiments on them to physical systems, unless appropriate heterogeneities are taken into account.

Congratulations to Dr. Alejandro Martínez!

My doctoral student Alejandro Martínez has now officially finished his D.Phil. (i.e., Ph.D.), with a dissertation called Disordered Granular Crystals.

His thesis work includes several awesome papers, including this one, this one, this one, this one, and this one. (Alejandro also has additional papers from his thesis era that are in collaboration with other people.)

Alejandro is now a postdoctoral scholar in computational biology in Chile.

"Quasiperiodic Granular Chains and Hofstadter Butterflies"

Our article just came out in final form today. It provides the cover picture of an issue of Philosophical Transactions of the Royal Society A. Here are some details.

Title: Quasiperiodic Granular Chains and Hofstadter Butterflies

Authors: Alejandro J. Martínez, Mason A. Porter, and Panayotis G. Kevrekidis

Abstract: We study quasiperiodicity-induced localization of waves in strongly precompressed granular chains. We propose three different set-ups, inspired by the Aubry–André (AA) model, of quasiperiodic chains; and we use these models to compare the effects of on-site and off-site quasiperiodicity in nonlinear lattices. When there is purely on-site quasiperiodicity, which we implement in two different ways, we show for a chain of spherical particles that there is a localization transition (as in the original AA model). However, we observe no localization transition in a chain of cylindrical particles in which we incorporate quasiperiodicity in the distribution of contact angles between adjacent cylinders by making the angle periodicity incommensurate with that of the chain. For each of our three models, we compute the Hofstadter spectrum and the associated Minkowski–Bouligand fractal dimension, and we demonstrate that the fractal dimension decreases as one approaches the localization transition (when it exists). We also show, using the chain of cylinders as an example, how to recover the Hofstadter spectrum from the system dynamics. Finally, in a suite of numerical computations, we demonstrate localization and also that there exist regimes of ballistic, superdiffusive, diffusive and subdiffusive transport. Our models provide a flexible set of systems to study quasiperiodicity-induced analogues of Anderson phenomena in granular chains that one can tune controllably from weakly to strongly nonlinear regimes.

This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

Apple Seismology

Here is a cool 'Quick Study' about "apple seismology".

Quoting the article's lead: Just as an earthquake’s seismic waves reveal properties of Earth’s interior, elastic surface waves on an apple can tell us about what’s going on inside the fruit.

This research may be a contender for an Ig Nobel prize.

Update: Apple seismology was discussed originally in a 1973 mathematical modeling paper by J.R. Cooke and Richard Rand. (This paper is cited in the Physics Today article above.)

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.

Playing "Soliton" in "Scrabble" (Lexulous)

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

"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.

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. :)

"Nonlinear Coherent Structures in Granular Crystals"

My review article on granular crystals (and especially work on it during the last decade) just came out in final form. Here are the details.

Title: Nonlinear Coherent Structures in Granular Crystals

Authors: Chris Chong, Mason A. Porter, Panayotis G. Kevrekidis, and Chiara Daraio

Abstract: The study of granular crystals, which are nonlinear metamaterials that consist of closely packed arrays of particles that interact elastically, is a vibrant area of research that combines ideas from disciplines such as materials science, nonlinear dynamics, and condensed-matter physics. Granular crystals exploit geometrical nonlinearities in their constitutive microstructure to produce properties (such as tunability and energy localization) that are not conventional to engineering materials and linear devices. In this topical review, we focus on recent experimental, computational, and theoretical results on nonlinear coherent structures in granular crystals. Such structures—which include traveling solitary waves, dispersive shock waves, and discrete breathers—have fascinating dynamics, including a diversity of both transient features and robust, long-lived patterns that emerge from broad classes of initial data. In our review, we primarily discuss phenomena in one-dimensional crystals, as most research to date has focused on such scenarios, but we also present some extensions to two-dimensional settings. Throughout the review, we highlight open problems and discuss a variety of potential
engineering applications that arise from the rich dynamic response of granular crystals.

Popsicle-Stick "Cobra Wave"

This is cool! Here is an explanation.

The cobra wave is a physical phenomenon arising from the explosion of a metastable grillage made of popsicle stickshttps://t.co/ISGSN8GCyY pic.twitter.com/5iFhkGgANC

— Physical Review Lett (@PhysRevLett) August 25, 2017

We got the video from PRL's tweet.

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

"Scattering of Waves by Impurities in Precompressed Granular Chains"

You may have heard of the Ramsaur–Townsend resonance from scattering problems. The simplest version of it (by examing a particle in a square well) is one of the canonical textbook problems in quantum mechanics.

This is one of the important effects illustrating the need for a notion of wave mechanics.

It turns out that one can also get an RT resonance in a macroscopic, classical system.

The discovery of this classical RT effect (in granular crystals) is the subject of a paper by my collaborators and me, out in final form in Physical Review E today. Here are the details of the article.


Title: "Scattering of Waves by Impurities in Precompressed Granular Chains"

Authors: Alejandro J. Martínez, Hiromi Yasuda, Eunho Kim, P. G. Kevrekidis, Mason A. Porter, and Jinkyu Yang

Abstract: We study scattering of waves by impurities in strongly precompressed granular chains. We explore the linear scattering of plane waves and identify a closed-form expression for the reflection and transmission coefficients for the scattering of the waves from both a single impurity and a double impurity. For single-impurity chains, we show that, within the transmission band of the host granular chain, high-frequency waves are strongly attenuated (such that the transmission coefficient vanishes as the wavenumber k → ±π), whereas low-frequency waves are well-transmitted through the impurity. For double-impurity chains, we identify a resonance—enabling full transmission at a particular frequency—in a manner that is analogous to the Ramsauer–Townsend (RT) resonance from quantum physics. We also demonstrate that one can tune the frequency of the RT resonance to any value in the pass band of the host chain. We corroborate our theoretical predictions both numerically and experimentally, and we directly observe almost complete transmission for frequencies close to the RT resonance frequency. Finally, we show how this RT resonance can lead to the existence of reflectionless modes in granular chains (including disordered ones) with multiple double impurities.

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.

"Superdiffusive Transport and Energy Localization in Disordered Granular Crystals"

A new paper of mine just came out in final form today. Here are the details. (Also see a paper by others published as a consecutive article with ours. Scientifically, it's really good that these articles have appeared as back-to-back papers.)

Title: Superdiffusive Transport and Energy Localization in Disordered Granular Crystals

Authors: Alejandro J. Martínez, P. G. Kevrekidis, and Mason A. Porter

Abstract: We study the spreading of initially localized excitations in one-dimensional disordered granular crystals. We thereby investigate localization phenomena in strongly nonlinear systems, which we demonstrate to differ fundamentally from localization in linear and weakly nonlinear systems. We conduct a thorough comparison of wave dynamics in chains with three different types of disorder—an uncorrelated (Anderson-like) disorder and two types of correlated disorders (which are produced by random dimer arrangements)—and for two types of initial conditions (displacement excitations and velocity excitations). We find for strongly precompressed (i.e., weakly nonlinear) chains that the dynamics depend strongly on the type of initial condition. In particular, for displacement excitations, the long-time asymptotic behavior of the second moment ˜m2 of the energy has oscillations that depend on the type of disorder, with a complex trend that differs markedly from a power law and
which is particularly evident for an Anderson-like disorder. By contrast, for velocity excitations, we find that a standard scaling m_2 ∼ t^γ (for some constant γ) applies for all three types of disorder. For weakly precompressed (i.e., strongly nonlinear) chains, m_2 and the inverse participation ratio P^{−1} satisfy scaling relations m_2 ∼ t^γ and P^{−1} ∼ t^{−η}, and the dynamics is superdiffusive for all of the cases that we consider. Additionally, when precompression is strong, the inverse participation ratio decreases slowly (with η < 0.1) for all three types of disorder, and the dynamics leads to a partial localization around the core and the leading edge of a propagating wave packet. For an Anderson-like disorder, displacement perturbations lead to localization of energy primarily in the core, and velocity perturbations cause the energy to be divided between the core and the leading edge. This localization phenomenon does not occur in the sonic-vacuum regime, which yields the surprising result that the energy is no longer contained in strongly nonlinear waves but instead is spread across many sites. In this regime, the exponents are very similar (roughly γ ≈ 1.7 and η ≈ 1) for all three types of disorder and for both types of initial conditions.