Energy Criterion for the Spectral Stability of Discrete Breathers

Kevrekidis, P. G.; Cuevas–Maraver, Jesús; Pelinovsky, Dmitry E.

doi:10.1103/physrevlett.117.094101

Cited by 28 publications

(33 citation statements)

References 37 publications

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“…We note that boundary effects are significant in the presence of oscillating tails, so the finite-length DBs do not accurately approximate the case of an infinite lattice. As an aside, we also point out that contrary, e.g., to what is the case in the work of [27,29] for a granular crystal, here the dependence of the energy (Hamiltonian) on the frequency is monotonic, hence in accordance with the recent criterion of [48], no instability arises from changes of monotonicity in this dependence.…”

Section: Numerical Investigation Of Breatherssupporting

confidence: 62%

Discrete breathers in a mass-in-mass chain with Hertzian local resonators

et al. 2017

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We report on the existence of discrete breathers in a one-dimensional, mass-in-mass chain with linear intersite coupling and nonlinear, precompressed Hertzian local resonators, which is motivated by recent studies of the dynamics of microspheres adhered to elastic substrates. After predicting theoretically the existence of discrete breathers in the continuum and anticontinuum limits of intersite coupling, we use numerical continuation to compute a family of breathers interpolating between the two regimes in a finite chain, where the displacement profiles of the breathers are localized around one lattice site. We then analyze the frequency-amplitude dependence of the breathers by performing numerical continuation on a linear eigenmode (vanishing amplitude) solution of the system near the upper band gap edge. Finally, we use direct numerical integration of the equations of motion to demonstrate the formation and evolution of the identified localized modes in energy-conserving and dissipative scenarios, including within settings that may be relevant to future experimental studies.

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Section: Numerical Investigation Of Breatherssupporting

confidence: 62%

Discrete breathers in a mass-in-mass chain with Hertzian local resonators

et al. 2017

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“…Here the effective frequency ω is proportional to their velocity c according to ω = 2πc/h. This, in turn, directly connects the criterion we analyze with a recently established criterion for the spectral stability of discrete breathers [16]. We emphasize here that the unifying connection of STWs with breathers does not impose any a priori restrictions on the nature of their decay of at infinity.…”

Section: Introductionmentioning

confidence: 97%

“…(2) Given the intimate connection of lattice STWs and DBs, an immediate correlation emerges between the criteria for stability change of discrete breathers, such as H (ω) = 0 that was recently established in Ref. [16] and the stability of lattice STWs discussed here (and also in Ref. [10]).…”

Section: A Complementary Perspective: Floquet Analysis Of the Timentioning

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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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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“…. We now turn to the constraint (A7), which again holds for any c. Expanding both sides in ǫ and using (20), (21), we obtain…”

Section: B Reduced Eigenvalue Problem With Constraints and The Leadimentioning

confidence: 99%

“…, Ω kk . Once all diagonal terms are known, given Ω 10 , one can solve for Ω 21 , Ω 32 , ..., Ω k(k−1) . Assuming Ω ji are known for any j − i < m and given Ω m0 , one can then obtain Ω (m+1) 1 , Ω (m+2)2 , .…”

Section: Proof Of Proposition 42mentioning

confidence: 99%

An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices

Cuevas–Maraver

Kevrekidis

et al. 2018

Phil. Trans. R. Soc. A.

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In this work, we revisit a criterion, originally proposed in Friesecke & Pego (Friesecke & Pego 2004 , 207-227. (doi:10.1088/0951715/17/1/013)), for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

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Energy Criterion for the Spectral Stability of Discrete Breathers

Cited by 28 publications

References 37 publications

Discrete breathers in a mass-in-mass chain with Hertzian local resonators

Discrete breathers in a mass-in-mass chain with Hertzian local resonators

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

An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices

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