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

Xu, Haitao; Cuevas–Maraver, Jesús; Kevrekidis, P. G.; Vainchtein, Anna

doi:10.1098/rsta.2017.0192

Cited by 18 publications

(31 citation statements)

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“…The energy criterion has been successfully applied for predicting stability changes in travelling waves (supersonic lattice solitons) in FPUT lattices with soft potentials [78,79]. In particular, the connection between breathers and traveling waves is an intriguing one: traveling waves on a lattice typically return to themselves upon translation by a single lattice site.…”

Section: Linear Stability At Arbitrary Coupling An Energy-based Stabi...mentioning

confidence: 99%

Discrete Breathers in $$\phi ^4$$ and Related Models

Cuevas–Maraver

Kevrekidis

2019

Nonlinear Systems and Complexity

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In this Chapter, we touch upon the wide topic of discrete breather (DB) formation with a special emphasis on the prototypical system of interest, namely the φ 4 model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the DBs. Our main emphasis is on the existence, and especially on the stability features of such solutions. We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy vs. frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted. While this Chapter by no means aspires to be comprehensive, we hope that it provides some recent developments (a large fraction of which is not included in time-honored DB reviews) and associated future possibilities.

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Section: Linear Stability At Arbitrary Coupling An Energy-based Stabi...mentioning

confidence: 99%

Discrete Breathers in $$\phi ^4$$ and Related Models

Cuevas–Maraver

Kevrekidis

2019

Nonlinear Systems and Complexity

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“…If ω is a frequency of the discrete breathers and H is the value of their energy, then the monotonicity of the dependence ω → H is related to the monotonicity of the dependence Ω → ν in the dNLS limit. Further results on the energy criterion for spectral stability of discrete breathers in the dKG equation (1.1) are given in [12,27]. In spite of many convincing numerical evidences, the orbital stability of single-pulse discrete breathers is still out of reach in the energy methods.…”

Section: Introductionmentioning

confidence: 99%

Existence, linear stability and long-time nonlinear stability of Klein-Gordon breathers in the small-amplitude limit

Pelinovsky,

Penati,

Paleari

2019

Preprint

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In this paper we consider a discrete Klein-Gordon (dKG) equation on Z d in the limit of the discrete nonlinear Schrödinger (dNLS) equation, for which small-amplitude breathers have precise scaling with respect to the small coupling strength ǫ. By using the classical Lyapunov-Schmidt method, we show existence and linear stability of the KG breather from existence and linear stability of the corresponding dNLS soliton. Nonlinear stability, for an exponentially long time scale of the order O(exp(ǫ −1 )), is also obtained via the normal form technique, together with higher order approximations of the KG breather through perturbations of the corresponding dNLS soliton.

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“…This is because the spectrum of the Lyapunov functional can be related to the spectrum of the operator arising when linearizing the governing PDE [27,28]. In this issue, two contributions [29,30] deal with the topic using the Hamiltonian structure to study spectral stability.…”

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confidence: 99%

“…Xu et al [29] consider travelling wave solutions to lattice Hamiltonian systems and study a condition for a pair of eigenvalues associated with the PDE system to cross zero and emerge on the real axis, thus leading to spectral instability. The authors perform their study following two different approaches: one based on Floquet multipliers, and one that is Hamiltonian-based.…”

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confidence: 99%

“…The authors perform their study following two different approaches: one based on Floquet multipliers, and one that is Hamiltonian-based. The article [29] and a previous one from the same authors [31] distinguish themselves from other works on lattice equations by the fact that no tools from integrability are used for the stability analysis. Two examples of lattice dynamical system problems that have solitary wave solutions that change stability are considered.…”

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Stability of nonlinear waves and patterns and related topics

Ghazaryan

Lafortune

Manukian

2018

Phil. Trans. R. Soc. A.

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Periodic and localized travelling waves such as wave trains, pulses, fronts and patterns of more complex structure often occur in natural and experimentally built systems. In mathematics, these objects are realized as solutions of nonlinear partial differential equations. The existence, dynamic properties and bifurcations of those solutions are of interest. In particular, their stability is important for applications, as the waves that are observable are usually stable. When the waves are unstable, further investigation is warranted of the way the instability is exhibited, i.e. the nature of the instability, and also coherent structures that appear as a result of an instability of travelling waves. A variety of analytical, numerical and hybrid techniques are used to study travelling waves and their properties.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.

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

Cited by 18 publications

References 56 publications

Discrete Breathers in $$\phi ^4$$ and Related Models

Discrete Breathers in $$\phi ^4$$ and Related Models

Existence, linear stability and long-time nonlinear stability of Klein-Gordon breathers in the small-amplitude limit

Stability of nonlinear waves and patterns and related topics

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