Discrete breathers in a two-dimensional Fermi–Pasta–Ulam lattice

Butt, Imran A; Wattis, Jonathan A. D.

doi:10.1088/0305-4470/39/18/013

Cited by 76 publications

(68 citation statements)

References 45 publications

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“…Switching the interaction between the equations back on, it would be a miracle if all equations now support moving pulses with one and the same velocity. Many further numerical and analytical investigations have shown since, that whatever small, oscillatory tails are typically unavoidable when constructing moving breather states [264,6,251,34,151,248,168,370,53,408].…”

Section: A Numerical Methods and Analytical Consequencesmentioning

confidence: 99%

Discrete breathers — Advances in theory and applications

Flach¹,

Gorbach²

2008

Physics Reports

941

830

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Section: A Numerical Methods and Analytical Consequencesmentioning

confidence: 99%

Discrete breathers — Advances in theory and applications

Flach¹,

Gorbach²

2008

Physics Reports

941

830

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“…It is natural to consider the relationship between the honeycomb system studied here and a onedimensional systems. We note that the two-dimensional systems studied previously [6,7] both had a dispersion relation with a single branch that described modes with optical and acoustic characters, as in one-dimensional (monatomic) systems. However, in one-dimensional diatomic systems, the dispersion relation has two branches, one optical and one acoustic.…”

Section: Discussionmentioning

confidence: 86%

“…Square [6] Hexagonal [7] Honeycomb Second harmonic The absence of any second harmonic is a property shared with the square lattice. Whilst the hexagonal lattice generates no third harmonic, it does generate a second harmonic.…”

Section: Property \ Geometrymentioning

confidence: 99%

Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices

Wattis

James²

2014

J. Phys. A: Math. Theor.

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We consider the two-dimensional Fermi-Pasta-Ulam lattice with hexagonal honeycomb symmetry, which is a Hamiltonian system describing the evolution of a scalar-valued quantity subject to nearest neighbour interactions. Using multiple-scale analysis we reduce the governing lattice equations to a nonlinear Schrödinger (NLS) equation coupled to a second equation for an accompanying slow mode. Two cases in which the latter equation can be solved and so the system decoupled are considered in more detail: firstly, in the case of a symmetric potential, we derive the form of moving breathers. We find an ellipticity criterion for the wavenumbers of the carrier wave, together with asymptotic estimates for the breather energy. The minimum energy threshold depends on the wavenumber of the breather. We find that this threshold is locally maximised by stationary breathers. Secondly, for an asymmetric potential we find stationary breathers, which, even with a quadratic nonlinearity generate no second harmonic component in the breather. Plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure. Finally, we compare the properties of stationary breathers in the square, triangular and honeycomb lattices.

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“…For example, it has been used for describing transversal mechanical vibrations of plane lattice in [6] (see Fig. 1), charge vibrations in an electrical network of nonlinear capacitors coupled to each other with linear inductors in [11,13] (see Fig. 2), etc.…”

Section: Introductionmentioning

confidence: 99%

Discrete breathers on symmetry-determined invariant manifolds for scalar models on the plane square lattice

2011

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A group-theoretical approach for studying localized periodic and quasiperiodic vibrations in two- and three-dimensional lattice dynamical models is developed. This approach is demonstrated for the scalar models on the plane square lattice. The symmetry-determined invariant manifolds admitting existence of localized vibrations are found, and some types of discrete breathers are constructed on these manifolds. A general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed. This method allows one to decompose the corresponding system of linear differential equations with time-dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system.

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Discrete breathers in a two-dimensional Fermi–Pasta–Ulam lattice

Cited by 76 publications

References 45 publications

Discrete breathers — Advances in theory and applications

Discrete breathers — Advances in theory and applications

Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices

Discrete breathers on symmetry-determined invariant manifolds for scalar models on the plane square lattice

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