Approximation of Polyatomic FPU Lattices by KdV Equations

Gaison, Jeremy; Moskow, Shari; Wright, James; Zhang, Qimin

doi:10.1137/130941638

Cited by 52 publications

(90 citation statements)

References 20 publications

(40 reference statements)

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“…This Fredholm orthogonality condition yields ε such that the solution (16) has no oscillations at infinity, and hence no such waves appear in q n (t) up to O(ε 2 ). Physically, it means that the slow motion of the center of mass of the two neighboring heavy masses, the acceleration of which equals −f 0 (t) up to a time shift, does not excite any fast oscillations of the light mass in between at large time.…”

Section: Asymptotic Analysismentioning

confidence: 99%

“…However, since g(ε; κ) → 0 as κ → 0, the amplitude of the trailing oscillations becomes exponentially small at κ near zero [50], which explains why quasicontinuum [17] and KdV [16] approximations of solitary waves for any ε work well in this regime.…”

Section: Asymptotic Analysismentioning

confidence: 99%

“…Using a quasicontinuum approximation, the authors obtained different solutions for soliton-like excitations, including subsonic and supersonic acoustic kinks and optical envelope solitons. Rigorous approximation results along these lines were established in [14][15][16]. Quasicontinuum approximations were also used to analyze solitary-like excitations and traveling kinks in diatomic lattices in [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

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Solitary waves in diatomic chains

et al. 2016

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We consider the mechanism of formation of isolated localized wave structures in the diatomic Fermi-Pasta-Ulam (FPU) model. Using a singular multiscale asymptotic analysis in the limit of high mass mismatch between the alternating elements, we obtain the typical slow-fast time scale separation and formulate the Fredholm orthogonality condition approximating a sequence of mass ratios supporting the formation of solitary waves in the general type of diatomic FPU models. This condition is made explicit in the case of diatomic Toda lattice. Results of numerical integration of the full diatomic Toda lattice equations confirm the formation of these genuinely localized wave structures at special values of the mass ratio that are close to the analytical predictions when the ratio is sufficiently small.

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Section: Asymptotic Analysismentioning

confidence: 99%

Section: Asymptotic Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solitary waves in diatomic chains

et al. 2016

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“…Gaison et al [21] apply homogenization techniques to the long wavelength limit of a nonlinear monoatomic lattice with periodic material properties. They uncover solutions that approximately satisfy a KdV equation.…”

mentioning

confidence: 99%

“…Friesecke and Pego [22] present that sonic speed wave pulses in nonlinear lattices are governed by the continuum limit regardless of the length scale. The studies in [20][21][22] provide insight into how invariant waveforms may arise in these lattices in the long wavelength limit.…”

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confidence: 99%

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

Fronk

Leamy

2017

Journal of Vibration and Acoustics

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Recent studies have presented first-order multiple time scale approaches for exploring amplitudedependent plane-wave dispersion in weakly nonlinear chains and lattices characterized by cubic stiffness. These analyses have yet to assess solution stability, which requires an analysis incorporating damping. Furthermore, due to their first-order dependence, they make an implicit assumption that cubic stiffness influences dispersion shifts to a greater degree than quadratic stiffness, and they thus ignore quadratic shifts. This paper addresses these limitations by carryingout higher-order, multiple scales perturbation analyses of linearly-damped nonlinear monoatomic and diatomic chains. The study derives higher-order dispersion corrections informed by both quadratic and cubic stiffness, and quantifies plane wave stability using evolution equations resulting from the multiple scales analysis and numerical experiments. Additionally, by reconstructing plane waves using both homogeneous and particular solutions at multiple orders, the study introduces a new interpretation of multiple scales results in which predicted waveforms are seen to exist over all space and time, constituting an invariant, multi-harmonic wave of infinite A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e dDownloaded From: http://vibrationacoustics.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jvacek/0/ on 04/24/2017 Terms of Use: http://www.asme.org/aboutVIB-16-1265, Leamy 2 extent analogous to cnoidal waves in continuous systems. Using example chains characterized by dimensionless parameters, numerical studies confirm that the spectral content of the predicted waveforms exhibits less growth/decay over time as higher-order approximations are used in defining the simulations' initial conditions. Thus the study results suggest that higher-order multiple scales perturbation analysis captures long-term, non-localized invariant plane waves, which have the potential for propagating coherent information over long distances.

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On the Korteweg‐de Vries approximation for a Boussinesq equation posed on the infinite necklace graph

Düll,

Schneider,

Taraca

2024

Math Methods in App Sciences

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Motivated by the question how to describe long‐wave dynamics on periodic networks, we consider a Boussinesq equation posed on the infinite periodic necklace graph. For the description of long‐wave traveling waves, we derive the KdV equation and establish the validity of this formal approximation by providing estimates for the error. The proof is based on suitable energy estimates. As a consequence of the approximation result, the soliton dynamics present in the KdV equation can approximately be seen for the original system, too. There are no serious obstacles to transfer the analysis to other dispersive systems or to other periodic quantum graphs for which the KdV equation can be derived. However, a general KdV validity theory on quantum graphs would be extremely abstract and of minor use.

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Approximation of Polyatomic FPU Lattices by KdV Equations

Cited by 52 publications

References 20 publications

Solitary waves in diatomic chains

Solitary waves in diatomic chains

Higher-Order Dispersion, Stability, and Waveform Invariance in Nonlinear Monoatomic and Diatomic Systems

On the Korteweg‐de Vries approximation for a Boussinesq equation posed on the infinite necklace graph

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