Fermi-Pasta-Ulam-Tsingou Research Articles

Scaling relations for auxin waves

We analyze an ‘up-the-gradient’ model for the formation of transport channels of the phytohormone auxin, through auxin-mediated polarization of the PIN1 auxin transporter. We show that this model admits a family of travelling wave solutions that is parameterized by the height of the auxin-pulse. We uncover scaling relations for the speed and width of these waves and verify these rigorous results with numerical computations. In addition, we provide explicit expressions for the leading-order wave profiles, which allows the influence of the biological parameters in the problem to be readily identified. Our proofs are based on a generalization of the scaling principle developed by Friesecke and Pego to construct pulse solutions to the classic Fermi–Pasta–Ulam–Tsingou model, which describes a one-dimensional chain of coupled nonlinear springs.

Fermi–Pasta–Ulam–Tsingou recurrence in two-core optical fibers

The Fermi–Pasta–Ulam–Tsingou recurrence (FPUT) refers to the property of a multi-mode nonlinear system to return to the initial states after complex stages of evolution. FPUT in two-core optical fibers (TCFs) is studied by taking symmetric continuous waves as initial conditions. Analytically the dynamics is governed by linearly coupled nonlinear Schrödinger equations. A ‘cascading mechanism’ is investigated theoretically to elucidate the FPUT dynamics. Higher-order modes grow in ‘locked step’ with the fundamental one until a breather emerges, which results in a triangular spectrum, in consistency with observations in many experiments. In terms of optical physics, FPUT with in-phase perturbations is identical to that in single-core fibers (SCFs). For quadrature-phase perturbations (those with π/2 phase shifts between the two cores), symmetry is broken and the FPUT dynamics is rich. For normal dispersion, FPUT otherwise absent in SCFs arises for TCFs. The optical fields change from a phase-shifted FPUT pattern to an ‘in-phase’ pattern. The number of FPUT cycles eventually saturates. For anomalous dispersion, in-phase FPUT and ‘repulsion’ between wave trains in the two cores are observed. Preservation of the phase difference between waves in the two cores depends on the dispersion regime. An estimate based on modulation instability provides a formula for predicting the first occurrence of FPUT. These results enhance the understanding of the physics of TCFs, and help the evaluation of the performance of long-distance communication based on multi-core fibers.

Analysis of ballistic transport and resonance in the α-Fermi-Pasta-Ulam-Tsingou model.

Ballistic transport and resonance phenomena are elucidated in the one-dimensional α-Fermi-Pasta-Ulam-Tsingou (FPUT) model using an approach of computing thermal response functions. The existence of periodic oscillations in spatially sinusoidal temperature profiles seen in previous studies is confirmed. However, the results obtained using response functions enable a more complete understanding. In particular, it is shown that resonance involves beats between normal modes which tend to reinforce in a one-dimensional chain. Anharmonic scattering acts to destroy phase coherence across the statistical ensemble, and with increasing anharmonicity, transport is driven toward the diffusive regime. These results provide additional insight into anomalous heat transport in low-dimensional systems. Normal-mode scattering is also explored using time correlation functions. Interestingly, these calculations, in addition to demonstrating loss of phase coherence across an ensemble of simulations, appear to show evidence of so-called q-breathers in conditions of strong anharmonicity. Finally, we describe how the approach outlined here could be developed to include quantum statistics and also also first-principles estimates of phonon scattering rates to elucidate second sound and ballistic transport in realistic materials at low temperatures.

The Fermi–Pasta–Ulam–Tsingou recurrence for discrete systems: Cascading mechanism and machine learning for the Ablowitz–Ladik equation

The Fermi–Pasta–Ulam–Tsingou recurrence phenomenon for the Ablowitz–Ladik equation is studied analytically and computationally. Wave profiles periodic in the discrete coordinate may return to the initial states after complex stages of evolution. Theoretically this dynamics is interpreted through a cascading mechanism where higher order harmonics exponentially small initially grow at a faster rate than the fundamental mode. A breather is formed when all modes attain roughly the same magnitude. Numerically a fourth-order Runge–Kutta method is implemented to reproduce this recurring pattern. In another illuminating perspective, we employ data driven and machine learning techniques, e.g. back propagation, hidden physics and physics-informed neural networks. Using data from a fixed time as a learning basis, doubly periodic solutions in both the defocusing and focusing regimes are obtained. The predictions by neural networks are in excellent agreement with those from numerical simulations and analytical solutions.

Unsteady two-temperature heat transport in mass-in-mass chains.

We investigate the unsteady heat (energy) transport in an infinite mass-in-mass chain with a given initial temperature profile. The chain consists of two sublattices: the β-Fermi-Pasta-Ulam-Tsingou (FPUT) chain and oscillators (of a different mass) connected to each FPUT particle. Initial conditions are such that initial kinetic temperatures of the FPUT particles and the oscillators are equal. Using the harmonic theory, we analytically describe evolution of these two temperatures in the ballistic regime. In particular, we derive a closed-form fundamental solution and solution for a sinusoidal initial temperature profile in the case when the oscillators are significantly lighter than the FPUT particles. The harmonic theory predicts that during the heat transfer the temperatures of sublattices are significantly different, while initially and finally (at large times) they are equal. This may look like an artifact of the harmonic approximation, but we show that it is not the case. Two distinct temperatures are also observed in the anharmonic case, even when the heat transport regime is no longer quasiballistic. We show that the value of the nonlinearity coefficient required to equalize the temperatures strongly depends on the particle mass ratio. If the oscillators are much lighter than the FPUT particles, then a fairly strong nonlinearity is required for the equalization.

Long-term instability of the inner Solar system: numerical experiments

ABSTRACT Apart from being chaotic, the inner planets in the Solar system constitute an open system, as they are forced by the regular long-term motion of the outer ones. No integrals of motion can bound a priori the stochastic wanderings in their high-dimensional phase space. Still, the probability of a dynamical instability is remarkably low over the next 5 billion years, a time-scale 1000 times longer than the Lyapunov time. The dynamical half-life of Mercury has indeed been estimated recently at 40 billion years. By means of the computer algebra system trip, we consider a set of dynamical models resulting from truncation of the forced secular dynamics recently proposed for the inner planets at different degrees in eccentricities and inclinations. Through ensembles of 103–105 numerical integrations spanning 5–100 Gyr, we find that the Hamiltonian truncated at degree 4 practically does not allow any instability over 5 Gyr. The destabilization is mainly due to terms of degree 6. This surprising result suggests an analogy to the Fermi–Pasta–Ulam–Tsingou problem, in which tangency to Toda Hamiltonian explains the very long time-scale of thermalization, which Fermi unsuccessfully looked for.

The anti-Fermi–Pasta–Ulam–Tsingou problem in one-dimensional diatomic lattices

We study the thermalization dynamics of one-dimensional diatomic lattices (which represents the simplest system possessing multi-branch phonons), exemplified by the famous Fermi–Pasta–Ulam–Tsingou (FPUT)-β and the Toda models. Here we focus on how the system relaxes to the equilibrium state when part of highest-frequency optical modes are initially excited, which is called the anti-FPUT problem comparing with the original FPUT problem (low frequency excitations of the monatomic lattice). It is shown numerically that the final thermalization time T eq of the diatomic FPUT-β chain depends on whether its acoustic modes are thermalized, whereas the T eq of the diatomic Toda chain depends on the optical ones; in addition, the metastable state of both models have different energy distributions and lifetimes. Despite these differences, in the near-integrable region, the T eq of both models still follows the same scaling law, i.e. T eq is inversely proportional to the square of the perturbation strength. Finally, comparisons of the thermalization behavior between different models under various initial conditions are briefly summarized.

Wave-packet spreading in disordered soft architected structures.

We study the dynamical and chaotic behavior of a disordered one-dimensional elastic mechanical lattice, which supports translational and rotational waves. The model used in this work is motivated by the recent experimental results of Deng et al. [Nat. Commun. 9, 1 (2018)]. This lattice is characterized by strong geometrical nonlinearities and the coupling of two degrees-of-freedom (DoFs) per site. Although the linear limit of the structure consists of a linear Fermi-Pasta-Ulam-Tsingou lattice and a linear Klein-Gordon (KG) lattice whose DoFs are uncoupled, by using single site initial excitations on the rotational DoF, we evoke the nonlinear coupling between the system's translational and rotational DoFs. Our results reveal that such coupling induces rich wave-packet spreading behavior in the presence of strong disorder. In the weakly nonlinear regime, we observe energy spreading only due to the coupling of the two DoFs (per site), which is in contrast to what is known for KG lattices with a single DoF per lattice site, where the spreading occurs due to chaoticity. Additionally, for strong nonlinearities, we show that initially localized wave-packets attain near ballistic behavior in contrast to other known models. We also reveal persistent chaos during energy spreading, although its strength decreases in time as quantified by the evolution of the system's finite-time maximum Lyapunov exponent. Our results show that flexible, disordered, and strongly nonlinear lattices are a viable platform to study energy transport in combination with multiple DoFs (per site), also present an alternative way to control energy spreading in heterogeneous media.

Wave propagation with longitudinal dust grain oscillations in dusty plasma crystals

Traveling waves are studied with a longitudinal dust grain oscillation in dusty plasma crystals modeled by α,β-Fermi Pasta Ulam Tsingou equation. Within the phonon band of the dispersion relation, the quasidiscrete approximation is applied to the relative displacement of the model to derive the nonlinear Schrödinger equation. The bright soliton solution is used as an initial condition for the full integration. The propagation of the robust bright wave is observed for the frequency far from the cutoff frequency while the localize wave is obtained at the cutoff frequency. In the forbidden bandgap, an analytical amplitude threshold of the nonlinear supratransmission phenomenon is derived. The numerical simulation confirms the validity of the analytical threshold, and the train of bright soliton is generated. Finally, by harmonically driving the equation of the longitudinal dusty plasma crystal, the traveling asymmetric bright soliton is generated for the first time from the plane wave.

The Cutoff Phenomenon in Wasserstein Distance for Nonlinear Stable Langevin Systems with Small Lévy Noise

This article establishes the cutoff phenomenon in the Wasserstein distance for systems of nonlinear ordinary differential equations with a dissipative stable fixed point subject to small additive Markovian noise. This result generalizes the results shown in Barrera, Högele, Pardo (EJP2021) in a more restrictive setting of Blumenthal-Getoor index α>3/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >3/2$$\\end{document} to the formulation in Wasserstein distance, which allows to cover the case of general Lévy processes with some given moment. The main proof techniques are based on the close control of the errors in a version of the Hartman–Grobman theorem and the adaptation of the linear theory established in Barrera, Högele, Pardo (JSP2021). In particular, they rely on the precise asymptotics of the nonlinear flow and the nonstandard shift linearity property of the Wasserstein distance, which is established by the authors in (JSP2021). Main examples are the nonlinear Fermi–Pasta–Ulam–Tsingou gradient flow and dissipative nonlinear oscillators subject to small (and possibly degenerate) Brownian or arbitrary α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-stable noise.

Micropterons, nanopterons and solitary wave solutions to the diatomic Fermi–Pasta–Ulam–Tsingou problem

We use a specialized boundary-value problem solver for mixed-type functional differential equations to numerically examine the landscape of traveling wave solutions to the diatomic Fermi–Pasta–Ulam–Tsingou (FPUT) problem. By using a continuation approach, we are able to uncover the relationship between the branches of micropterons and nanopterons that have been rigorously constructed recently in various limiting regimes. We show that the associated surfaces are connected together in a nontrivial fashion and illustrate the key role that solitary waves play in the branch points. Finally, we numerically show that the diatomic solitary waves are stable under the full dynamics of the FPUT system.

Energy spreading, equipartition, and chaos in lattices with non-central forces

We numerically study a one-dimensional, nonlinear lattice model which in the linear limit is relevant to the study of bending (flexural) waves. In contrast with the classic one-dimensional mass-spring system, the linear dispersion relation of the considered model has different characteristics in the low frequency limit. By introducing disorder in the masses of the lattice particles, we investigate how different nonlinearities in the potential (cubic, quadratic, and their combination) lead to energy delocalization, equipartition, and chaotic dynamics. We excite the lattice using single site initial momentum excitations corresponding to a strongly localized linear mode and increase the initial energy of excitation. Beyond a certain energy threshold, when the cubic nonlinearity is present, the system is found to reach energy equipartition and total delocalization. On the other hand, when only the quartic nonlinearity is activated, the system remains localized and away from equipartition at least for the energies and evolution times considered here. However, for large enough energies for all types of nonlinearities we observe chaos. This chaotic behavior is combined with energy delocalization when cubic nonlinearities are present, while the appearance of only quadratic nonlinearity leads to energy localization. Our results reveal a rich dynamical behavior and show differences with the relevant Fermi–Pasta–Ulam–Tsingou model. Our findings pave the way for the study of models relevant to bending (flexural) waves in the presence of nonlinearity and disorder, anticipating different energy transport behaviors.

Measuring α-FPUT Cores and Tails

Almost 70 years ago, the Fermi–Pasta–Ulam–Tsingou (FPUT) paradox was formulated in, observed in, and reported using normal modes of a nonlinear, one-dimensional, non-integrable string. Let us recap the paradox. One normal mode is excited, which drives three or four more normal modes in the core. Then, that is it for quite a long time. So why are many normal modes staying weakly excited in the tail? Furthermore, how many? A quantitative, analytical answer to the latter question is given here using resonances and secular avalanches A comparison with the previous numerical data is made and extremely good agreement is found.

On the convergence of the nonlocal nonlinear model to the classical elasticity equation

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional propagation of nonlinear waves in a continuous medium. In the limit of vanishing nonlocality we study the behavior of solutions to the Cauchy problem. We prove that, as the kernel functions of the convolution integral approach the Dirac delta function, the solutions converge strongly to the corresponding solutions of the classical elasticity equation. An energy estimate with no loss of derivative plays a critical role in proving the convergence result. As a typical example, we consider the continuous limit of the discrete lattice dynamic model (the Fermi–Pasta–Ulam–Tsingou model) and show that, as the lattice spacing approaches zero, solutions to the discrete lattice equation converge to the corresponding solutions of the classical elasticity equation.

Quasi-stable localized excitations in the β-Fermi Pasta Ulam Tsingou system

The lifetimes of localized nonlinear modes in both the $\beta$-Fermi-Pasta-Ulam-Tsingou ($\beta$-FPUT) chain and a cubic $\beta$-FPUT lattice are studied as functions of perturbation amplitude, and by extension, the relative strength of the nonlinear interactions compared to the linear part. We first recover the well known result that localized nonlinear excitations (LNEs) produced by a bond squeeze can be reduced to an approximate two-frequency solution and then show that the nonlinear term in the potential can lead to the production of secondary frequencies within the phonon band. This can affect the stability and lifetime of the LNE by facilitating interactions between the LNE and a low energy acoustic background which can be regarded as "noise" in the system. In the one dimensional FPUT chain, the LNE is stabilized by low energy acoustic emissions at early times; in some cases allowing for lifetimes several orders of magnitude larger than the oscillation period. The longest lived LNEs are found to satisfy the parameter dependence $\mathcal{A}\sqrt{\beta}\approx1.1$ where $\beta$ is the relative nonlinear strength and $\mathcal{A}$ is the displacement amplitude of the center particles in the LNE. In the cubic FPUT lattice, the LNE lifetime $T$ decreases rapidly with increasing amplitude $\mathcal{A}$ and is well described by the double log relationship $\log_{10}\log_{10}(T)\approx -(0.15\pm0.01)\mathcal{A}\sqrt{\beta}+(0.62\pm0.02)$.

Four-wave mixing and coherently coupled Schrödinger equations: Cascading processes and Fermi-Pasta-Ulam-Tsingou recurrence.

Modulation instability, breather formation, and the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A "cascading mechanism" is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with "eye-shaped" breathers in one waveguide and "four-petal" modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.

Korteweg–de Vries and Fermi–Pasta–Ulam–Tsingou: asymptotic integrability of quasi unidirectional waves

In this paper we construct a higher order expansion of the manifold of quasi unidirectional waves in the Fermi–Pasta–Ulam–Tsingou (FPUT) chain. We also approximate the dynamics on this manifold. As perturbation parameter we use h 2 = 1/n 2, where n is the number of particles of the chain. It is well known that the dynamics of quasi unidirectional waves is described to first order by the Korteweg–de Vries (KdV) equation. Here we show that the dynamics to second order is governed by a combination of the first two nontrivial equations in the KdV hierarchy—for any choice of parameters in the FPUT potential. On the other hand, we find that only if the parameters of the FPUT potential satisfy a condition, then a combination of the first three nontrivial equations in the KdV hierarchy determines the dynamics of quasi unidirectional waves to third order. The required condition is satisfied by the Toda chain. Our results suggest why the close-to-integrable behavior of the FPUT chain (the FPUT paradox) persists on a time scale longer than explained by the KdV approximation, and also how a breakdown of integrability (detachment from the KdV hierarchy) may be responsible for the eventual thermalization of the system.

Triads and Rogue Events for Internal Waves in Stratified Fluids with a Constant Buoyancy Frequency

Internal waves in a stratified fluid with a constant buoyancy frequency were studied, with special attention given to rogue modes, extreme waves, dynamical evolution, and Fermi–Pasta–Ulam–Tsingou type recurrence phenomena. Rogue waves for triads in a general physical setting have recently been derived analytically, but the implications in a fluid mechanics context have not yet been fully assessed. Numerical simulations were conducted for cases of coupled triads where the common member is a daughter wave mode. In sharp contrast with previous studies, rogue modes instead of plane waves were used as the initial condition. Furthermore, spatial dependence was incorporated. Rogue or extreme waves in one set of triads provided a possible mechanism for significant energy transfer among modes of the internal wave spectrum, in addition to the other known theories, e.g., weak turbulence. Remarkably, Fermi–Pasta–Ulam–Tsingou recurrence types of growth and decay cycles arose, similar to those observed for surface gravity wave groups governed by the cubic nonlinear Schrödinger equation. These mechanisms will enhance our understanding of transport processes in oceans.

Nonlinearity accelerates the thermalization of the quartic FPUT model with stochastic baths

We investigate the equilibration process of the strongly coupled quartic Fermi–Pasta–Ulam–Tsingou model by adding Langevin baths to the ends of the chain. The time evolution of the system is investigated by means of extensive numerical simulations and shown to match the results expected from equilibrium statistical mechanics in the time-asymptotic limit. Upon increasing the nonlinear coupling, the thermalization of the energy spectrum displays an increasing asymmetry in favor of small-scale, high-frequency modes, which relax significantly faster than the large-scale, low-frequency ones. The global equilibration time is found to scale linearly with system size and shown to exhibit a power-law decay with the strength of the nonlinearity and temperature. Nonlinear interaction adds to energy distribution among modes, thus speeding up the thermalization process.

Nonlinear localized modes in two-dimensional hexagonally-packed magnetic lattices

We conduct an extensive study of nonlinear localized modes (NLMs), which are temporally periodic and spatially localized structures, in a two-dimensional array of repelling magnets. In our experiments, we arrange a lattice in a hexagonal configuration with a light-mass defect, and we harmonically drive the center of the chain with a tunable excitation frequency, amplitude, and angle. We use a damped, driven variant of a vector Fermi–Pasta–Ulam–Tsingou lattice to model our experimental setup. Despite the idealized nature of this model, we obtain good qualitative agreement between theory and experiments for a variety of dynamical behaviors. We find that the spatial decay is direction-dependent and that drive amplitudes along fundamental displacement axes lead to nonlinear resonant peaks in frequency continuations that are similar to those that occur in one-dimensional damped, driven lattices. However, we observe numerically that driving along other directions results in asymmetric NLMs that bifurcate from the main solution branch, which consists of symmetric NLMs. We also demonstrate both experimentally and numerically that solutions that appear to be time-quasiperiodic bifurcate from the branch of symmetric time-periodic NLMs.