arXiv subjects

Christopher Chong

Publications and source records attributed to Christopher Chong.

At least 19 recordsLinked to original sources

arXiv2026

Wedge problems and dispersive shock waves in the two-dimensional Toda lattice

We study the formation and interaction of dispersive shock waves (DSWs) in the two-dimensional Toda lattice subject to wedge-type initial conditions, and show that their interaction gives rise to a discrete analog of Mach reflection for dispersive shock waves in discrete systems. The initial jump across each leg of the wedge acts locally as a Riemann problem for the one-dimensional Toda lattice, producing two oblique DSWs whose leading-edge soliton amplitude is determined explicitly by the one-dimensional Whitham modulation theory. The two-dimensional nature of the problem manifests when these oblique DSWs meet along the symmetry axis. We show that, for compressive wedges (i.e., when the initial conditions are such that two oblique DSWs that are generated propagate toward each other), a critical slope $q_{\mathrm{cr}}$ separates two regimes: in the subcritical regime ($q q_{\mathrm{cr}}$) the interaction is ordinary and produces a localized peak whose amplitude is determined analytically. We also show qualitatively that a similar dichotomy between two regimes exists for expansive wedges (i.e., when the initial conditions are such that the two oblique DSWs propagate away from each other). We confirm all analytical predictions by comparing them with the results of direct numerical simulations. Finally, we show that the continuum limit of the result is consistent with the analogous theory for the Kadomtsev-Petviashvili equation, providing an independent validation of the analytical framework.

nlin.PS↗

arXiv2026

Traveling and Dispersive Shock Waves in a Two-Dimensional Fermi-Pasta-Ulam-Tsingou Lattice

In the present work we analyze traveling and dispersive shock waves of a two-dimensional Fermi-Pasta-Ulam-Tsingou lattice. In the first part of the paper, using variational techniques we prove the existence of both periodic and solitary traveling waves for convex potentials. In the case of unimodal profiles we are able to remove the assumption of convexity. The variational formulation also provides a natural algorithm for the numerical computation of traveling waves, which we use to explore both solitary and periodic traveling waves. The numerical computations are compared with analytical approximations based on the derivation of the KdV equation for quasi-one-dimensional propagation. In the second part of the paper, we focus on dispersive shock waves (DSWs), which are expanding modulated waves that connect states of different amplitude. In particular, we focus on line DSWs, which are constant along one direction and propagate in the direction orthogonal to which it is constant. Such solutions form when subject to quasi-one-dimensional jump initial data. We find that while the shape of the DSW depends on the direction of travel, properties such as the speed and amplitude do not. The systematic numerical study of the line~DSWs is then compared to those predicted by the KdV equation along the line of propagation. Key characteristics of the DSWs, such as the speeds of the trailing and leading edges, are investigated for various jump heights, yielding good agreement between simulation and KdV approximation in the limit of vanishing jump height. Finally, we apply the DSW fitting method to study the trailing and leading edge characteristics of the DSW, finding even better agreement to the numerics when compared to the KdV prediction. The KdV prediction and DSW fitting predictions agree in the limit of small jump height.

nlin.PS↗

arXiv2026

Formation of mechanical rogue waves

Rogue waves, characterized by their abrupt and extreme localization in space and time, have evolved from maritime folklore to subjects of intense study across diverse fields, from hydrodynamics and nonlinear optics to plasmas and condensed matter physics. In mechanical systems, however, experimental realization remains elusive despite theoretical and numerical predictions. This gap stems from the stringent requirements for controllable nonlinearity, the high-fidelity initialization of the system, and the necessity to overcome inherent energy dissipation. Here, we report the experimental formation of mechanical rogue waves in a precisely engineered one-dimensional metamaterial lattice with tailored nonlinearity and minimal dissipative losses. Using a precision electromagnetic release system, we prescribe initial strain profiles that trigger a transition from dispersive decay to extreme wave focusing. Our parametric analysis reveals that the emergence of these extreme events is strictly contingent upon a synergy between high nonlinearity and a broad spatial energy reservoir within the initial seed. Crucially, neither factor alone is sufficient to overcome dispersion and trigger the observed focusing. These findings establish a robust platform for studying transient nonlinear wave focusing phenomena in mechanical systems and offer insights for harnessing extreme wave localization for applications such as energy harvesting, waveguiding, and mechanical signal processing.

nlin.PS↗

arXiv2025

First-order continuum models for nonlinear dispersive waves in the granular crystal lattice

We derive and analyze, analytically and numerically, two first-order continuum models to approximate the nonlinear dynamics of granular crystal lattices, focusing specifically on solitary waves, periodic waves, and dispersive shock waves. The dispersive shock waves predicted by the two continuum models are studied using modulation theory, DSW fitting techniques, and direct numerical simulations. The PDE-based predictions show good agreement with the DSWs generated by the discrete model simulation of the granular lattice itself, even in cases where no precompression is present and the lattice is purely nonlinear. Such an effective description could prove useful for future, more analytically amenable approximations of the original lattice system.

nlin.PS↗

arXiv2025

Unveiling Solitonic Collisions in Mechanical Metamaterials

Interactions between solitary waves have been pivotal to understanding nonlinear phenomena across various disciplines. The dynamics of rarefaction solitary waves holds great potential, yet their fundamental characteristics and interactions remain only partially understood through experimental means in mechanical metamaterials. Previous studies highlighted their existence and proposed applications, such as waveguides, impact mitigation, and energy harvesting. Challenges, including energy dissipation and a lack of precise measurement techniques, have hindered deeper exploration, most notably of solitonic collisions. In this work, we provide a definitive platform for examining pure rarefaction solitons propagating through a strain-softening mechanical lattice, addressing these challenges. Employing a theoretical framework based on the Boussinesq approximation and multiple-scale analysis, we predict soliton behavior, including phase shifts resulting from head-on collisions. These theoretical insights are corroborated through numerical simulations and systematic experiments designed to generate and measure pure rarefaction solitons with high precision. Both symmetric and asymmetric collisions are examined, revealing practically elastic interaction behaviors and amplitude-dependent phase shifts. Furthermore, collision dynamics, such as speed and phase shifts during rarefaction soliton collisions, from the experimental results show agreement with theoretical and numerical models. These results validate our experimental platform and findings, underscoring the potential of mechanical rarefaction solitons as robust, controllable wave packets. This suggests a robust paradigm for exploring nonlinear wave interactions in mechanical systems, opening new application avenues in mechanical metamaterials, such as wave-based computing and advanced signal processing.

nlin.PS↗

arXiv2024

A regularized continuum model for traveling waves and dispersive shocks of the granular chain

In this paper we focus on a discrete physical model describing granular crystals, whose equations of motion can be described by a system of differential difference equations (DDEs). After revisiting earlier continuum approximations, we propose a regularized continuum model variant to approximate the discrete granular crystal model through a suitable partial differential equation (PDE). We then compute, both analytically and numerically, its traveling wave and periodic traveling wave solutions, in addition to its conservation laws. Next, using the periodic solutions, we describe quantitatively various features of the dispersive shock wave (DSW) by applying Whitham modulation theory and the DSW fitting method. Finally, we perform several sets of systematic numerical simulations to compare the corresponding DSW results with the theoretical predictions and illustrate that the continuum model provides a good approximation of the underlying discrete one.

nlin.PS↗

arXiv2024

Dynamics of Nonlinear Lattices

In this topical review we explore the dynamics of nonlinear lattices with a particular focus to Fermi-Pasta-Ulam-Tsingou type models that arise in the study of elastic media and, more specifically, granular crystals. We first revisit the workhorse of such lattices, namely traveling waves, both from a continuum, but also from a genuinely discrete perspective, both without and with a linear force component (induced by the so-called precompression). We then extend considerations to time-periodic states, examining dark breather structures in homogeneous crystals, as well as bright breathers in diatomic lattices. The last pattern that we consider extensively is the dispersive shock wave arising in the context of suitable Riemann (step) initial data. We show how the use of continuum (KdV) and discrete (Toda) integrable approximations can be used to get a first quantitative handle of the relevant waveforms. In all cases, theoretical analysis is accompanied by numerical computations and, where possible, by a recap and illustration of prototypical experimental results. We close the chapter by offering a number of ongoing and potential future directions and associated open problems in the field.

nlin.PS↗

arXiv2024

On the Existence of Generalized Breathers and Transition Fronts in Time-Periodic Nonlinear Lattices

We prove the existence of a class of time-localized and space-periodic breathers (called q-gap breathers) in nonlinear lattices with time-periodic coefficients. These q-gap breathers are the counterparts to the classical space-localized and time-periodic breathers found in space-periodic systems. Using normal form transformations, we establish rigorously the existence of such solutions with oscillating tails (in the time domain) that can be made arbitrarily small, but finite. Due to the presence of the oscillating tails, these solutions are coined generalized q-gap breathers. Using a multiple-scale analysis, we also derive a tractable amplitude equation that describes the dynamics of breathers in the limit of small amplitude. In the presence of damping, we demonstrate the existence of transition fronts that connect the trivial state to the time-periodic ones. The analytical results are corroborated by systematic numerical simulations.

nlin.PS↗

arXiv2024

Hydrodynamics of a Discrete Conservation Law

The Riemann problem for the discrete conservation law $2 \dot{u}_n + u^2_{n+1} - u^2_{n-1} = 0$ is classified using Whitham modulation theory, a quasi-continuum approximation, and numerical simulations. A surprisingly elaborate set of solutions to this simple discrete regularization of the inviscid Burgers' equation is obtained. In addition to discrete analogues of well-known dispersive hydrodynamic solutions -- rarefaction waves (RWs) and dispersive shock waves (DSWs) -- additional unsteady solution families and finite time blow-up are observed. Two solution types exhibit no known conservative continuum correlates: (i) a counterpropagating DSW and RW solution separated by a symmetric, stationary shock and (ii) an unsteady shock emitting two counter-propagating periodic wavetrains with the same frequency connected to a partial DSW or a RW. Another class of solutions called traveling DSWs, (iii), consists of a partial DSW connected to a traveling wave comprised of a periodic wavetrain with a rapid transition to a constant. Portions of solutions (ii) and (iii) are interpreted as shock solutions of the Whitham modulation equations.

nlin.PS↗

arXiv2023

On the Whitham modulation equations for the Toda lattice and the quantitative characterization of its dispersive shocks

The aim of this work is multifold. Firstly, it intends to present a complete, quantitative and self-contained description of the periodic traveling wave solutions and Whitham modulation equations for the Toda lattice, combining results from different previous works in the literature. Specifically, we connect the Whitham modulation equations and a detailed expression for the periodic traveling wave solutions of the Toda lattice. Along the way, some key details are filled in, such as the explicit expression of the characteristic speeds of the genus-one Toda-Whitham system. Secondly, we use these tools to obtain a detailed quantitative characterization of the dispersive shocks of the Toda system. Lastly, we validate the relevant analysis by performing a detailed comparison with direct numerical simulations.

nlin.SI↗

arXiv2023

Modulation Instability and Wavenumber Bandgap Breathers in a Time Layered Phononic Lattice

We demonstrate the existence of wavenumber bandgap (q-gap) breathers in a time-periodic phononic lattice. These breathers are localized in time and periodic in space, and are the counterparts to the classical breathers found in spatially-periodic systems. We derive an exact condition for modulation instability that leads to the opening of wavenumber bandgaps. The q-gap breathers become more narrow and larger in amplitude as the wavenumber goes further into the bandgap. In the presence of damping, these structures acquire a non-zero, oscillating tail. The experiment and model exhibit qualitative agreement.

nlin.PS↗

arXiv2022

Breathers in lattices with alternating strain-hardening and strain-softening interactions

This work focuses on the study of time-periodic solutions, including breathers, in a nonlinear lattice consisting of elements whose contacts alternate between strain-hardening and strain-softening. The existence, stability, and bifurcation structure of such solutions, as well as the system dynamics in the presence of damping and driving are studied systematically. It is found that the linear resonant peaks in the system bend toward the frequency gap in the presence of nonlinearity. The time-periodic solutions that lie within the frequency gap compare well to Hamiltonian breathers if the damping and driving are small. In the Hamiltonian limit of the problem, we use a multiple scale analysis to derive a Nonlinear Schr\"odinger (NLS) equation to construct both acoustic and optical breathers. The latter compare very well with the numerically obtained breathers in the Hamiltonian limit.

nlin.PS↗

arXiv2022

Dispersive Shock Waves in Lattices: A Dimension Reduction Approach

Dispersive shock waves (DSWs), which connect states of different amplitude via a modulated wave train, form generically in nonlinear dispersive media subjected to abrupt changes in state. The primary tool for the analytical study of DSWs is Whitham's modulation theory. While this framework has been successfully employed in many space-continuous settings to describe DSWs, the Whitham modulation equations are virtually intractable in most spatially discrete systems. In this article, we illustrate the relevance of the reduction of the DSW dynamics to a planar ODE in a broad class of lattice examples. Solutions of this low-dimensional ODE accurately describe the orbits of the DSW in self-similar coordinates and the local averages in a manner consistent with the modulation equations. We use data-driven and quasi-continuum approaches within the context of a discrete system of conservation laws to demonstrate how the underlying low dimensional structure of DSWs can be identified and analyzed. The connection of these results to Whitham modulation theory is also discussed.

nlin.PS↗

arXiv2021

Rogue and solitary waves in coupled phononic crystals

In this work we present an analytical and numerical study of rogue and solitary waves in a coupled one-dimensional nonlinear lattice that involves both axial and rotational degrees of freedom. Using a multiple-scale analysis we derive a system of coupled nonlinear Schr\"odinger type equations in order to approximate solitary waves and rogue waves of the coupled lattice model. Numerical simulations are found to agree with the analytical approximations. We also consider generic initialization data in the form of a Gaussian profile and observe that they can result in the spontaneous formation of rogue wave-like patterns in the lattice. The solitary and rogue waves in the lattice demonstrate both energy isolation and exchange between the axial and rotational degrees of freedom of the system. This suggests that the studied coupled lattice has the potential to be an efficient energy isolation, transfer, and focusing medium.

nlin.PS↗

arXiv2018

Origami-based impact mitigation via rarefaction solitary wave creation

The principles underlying the art of origami paper folding can be applied to design sophisticated metamaterials with unique mechanical properties. By exploiting the flat crease patterns that determine the dynamic folding and unfolding motion of origami, we are able to design an origami-based metamaterial that can form rarefaction solitary waves. Our analytical, numerical and experimental results demonstrate that this rarefaction solitary wave overtakes initial compressive strain waves, thereby causing the latter part of the structure to feel tension first instead of compression. This counter-intuitive dynamic mechanism can be used to create a highly efficient--yet reusable--impact mitigating system without relying on material damping, plasticity or fracture.

nlin.PS↗

arXiv2016

Shock and Rarefaction Waves in Generalized Hertzian Contact Models

In the present work motivated by generalized forms of the Hertzian dynamics associated with granular crystals, we consider the possibility of such models to give rise to both shock and rarefaction waves. Depending on the value $p$ of the nonlinearity exponent, we find that both of these possibilities are realizable. We use a quasi-continuum approximation of a generalized inviscid Burgers model in order to predict the solution profile up to times near the shock formation, as well as to estimate when it will occur. Beyond that time threshold, oscillations associated with the discrete nature of the underlying model emerge that cannot be captured by the quasi-continuum approximation. Our analytical characterization of the above features is complemented by systematic numerical computations.

cond-mat.mtrl-sci↗

arXiv2016

Wave mixing in coupled phononic crystals via a variable stiffness mechanism

We investigate wave mixing effects in a phononic crystal that couples the wave dynamics of two channels -- primary and control ones -- via a variable stiffness mechanism. We demonstrate analytically and numerically that the wave transmission in the primary channel can be manipulated by the control channel's signal. We show that the application of control waves allows the selection of a specific mode through the primary channel. We also demonstrate that the mixing of two wave modes is possible whereby a modulation effect is observed. A detailed study of the design parameters is also carried out to optimize the switching capabilities of the proposed system. Finally, we verify that the system can fulfill both switching and amplification functionalities, potentially enabling the realization of an acoustic transistor.

nlin.PS↗

arXiv2016

A mechanical autonomous stochastic heat engine

Stochastic heat engines are devices that generate work from random thermal motion using a small number of highly fluctuating degrees of freedom. Proposals for such devices have existed for more than a century and include the Maxwell demon and the Feynman ratchet. Only recently have they been demonstrated experimentally, using e.g., thermal cycles implemented in optical traps. However, the recent demonstrations of stochastic heat engines are nonautonomous, since they require an external control system that prescribes a heating and cooling cycle, and consume more energy than they produce. This Report presents a heat engine consisting of three coupled mechanical resonators (two ribbons and a cantilever) subject to a stochastic drive. The engine uses geometric nonlinearities in the resonating ribbons to autonomously convert a random excitation into a low-entropy, nonpassive oscillation of the cantilever. The engine presents the anomalous heat transport property of negative thermal conductivity, consisting in the ability to passively transfer energy from a cold reservoir to a hot reservoir.

cond-mat.stat-mech↗