On the solitary wave pulse in a chain of beads

English, J. M.; Pego, Robert L.

doi:10.1090/s0002-9939-05-07851-2

Cited by 70 publications

(88 citation statements)

References 7 publications

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“…In this case, too, as identified via the methods of Refs. [6][7][8]22] (see Appendix B for details on numerical simulations) and shown in Fig. 1, the family of STWs numerically features H (c) > 0, in full agreement with their identification as stable.…”

Section: Numerical Corroborationsupporting

confidence: 73%

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Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

et al. 2017

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In this work, we provide two complementary perspectives for the (spectral) stability of solitary traveling waves in Hamiltonian nonlinear dynamical lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical examples. One is as an eigenvalue problem for a stationary solution in a cotraveling frame, while the other is as a periodic orbit modulo shifts. We connect the eigenvalues of the former with the Floquet multipliers of the latter and using this formulation derive an energy-based spectral stability criterion. It states that a sufficient (but not necessary) condition for a change in the wave stability occurs when the functional dependence of the energy (Hamiltonian) H of the model on the wave velocity c changes its monotonicity. Moreover, near the critical velocity where the change of stability occurs, we provide an explicit leading-order computation of the unstable eigenvalues, based on the second derivative of the Hamiltonian H (c 0 ) evaluated at the critical velocity c 0 . We corroborate this conclusion with a series of analytically and numerically tractable examples and discuss its parallels with a recent energy-based criterion for the stability of discrete breathers.

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Section: Numerical Corroborationsupporting

confidence: 73%

“…(9) of the FPU case but add long-range interactions with the kernel in Eq. (8). Figure 2 showcases the power of the stability criterion and illustrates the complementary nature of the cotraveling steady state and the periodic orbit FM calculation approaches.…”

Section: Numerical Corroborationmentioning

confidence: 99%

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

et al. 2017

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“…Note that these (Gaussian) homoclinic solutions decay super-exponentially, but do not decay doubly exponentially unlike the solitary wave solutions of the differential advance-delay equation (2.4) [30,31]. Homoclinic solution (3.16) yields an approximate Gaussian solitary wave solution of FPU lattice (1.8) with velocity equal to unity Ahnert & Pikovsky [28], based on a reformulation of (2.4) as a nonlinear integral equation and the method of successive approximations (see also [9,31] for variants of this method).…”

Section: (B) Stationary Solutionsmentioning

confidence: 99%

Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials

James

Pelinovsky

2014

Proc. R. Soc. A.

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We consider a class of fully nonlinear FermiPasta-Ulam (FPU) lattices, consisting of a chain of particles coupled by fractional power nonlinearities of order α > 1. This class of systems incorporates a classical Hertzian model describing acoustic wave propagation in chains of touching beads in the absence of precompression. We analyse the propagation of localized waves when α is close to unity. Solutions varying slowly in space and time are searched with an appropriate scaling, and two asymptotic models of the chain of particles are derived consistently. The first one is a logarithmic Korteweg-de Vries (KdV) equation and possesses linearly orbitally stable Gaussian solitary wave solutions. The second model consists of a generalized KdV equation with Hölder-continuous fractional power nonlinearity and admits compacton solutions, i.e. solitary waves with compact support. When α → 1 + , we numerically establish the asymptotically Gaussian shape of exact FPU solitary waves with near-sonic speed and analytically check the pointwise convergence of compactons towards the limiting Gaussian profile.

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“…We will seek them directly as traveling waves in real space, by attempting to identify fixed points of a discretization of the co-traveling frame nonlinear problem in our direct method. We will also follow the approach of [43], rewriting the problem in Fourier space (upon a Fourier transform) and seeking fixed points of that variant upon inverse Fourier transform, similarly to [39,43]. This will recover a convolution based reformulation of the problem in real space.…”

Section: Introductionmentioning

confidence: 99%

Traveling waves and their tails in locally resonant granular systems

Kevrekidis

Stefanov

2015

J. Phys. A: Math. Theor.

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In the present study, we revisit the theme of wave propagation in locally resonant granular crystal systems, also referred to as mass-in-mass systems. We use three distinct approaches to identify relevant traveling waves. The first consists of a direct solution of the traveling wave problem. The second one consists of the solution of the Fourier tranformed variant of the problem, or, more precisely, of its convolution reformulation (upon an inverse Fourier transform) in real space. Finally, our third approach will restrict considerations to a finite domain, utilizing the notion of Fourier series for important technical reasons, namely the avoidance of resonances, which will be discussed in detail. All three approaches can be utilized in either the displacement or the strain formulation. Typical resulting computations in finite domains result in the solitary waves bearing symmetric non-vanishing tails at both ends of the computational domain. Importantly, however, a countably infinite set of antiresonance conditions is identified for which solutions with genuinely rapidly decaying tails arise.

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On the solitary wave pulse in a chain of beads

Cited by 70 publications

References 7 publications

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

Gaussian solitary waves and compactons in Fermi–Pasta–Ulam lattices with Hertzian potentials

Traveling waves and their tails in locally resonant granular systems

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