Existence of breathers on FPU lattices

James, Guillaume

doi:10.1016/s0764-4442(01)01894-8

Cited by 56 publications

(60 citation statements)

References 13 publications

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“…To end this section we point out that recently a proof of existence of DB in FPU chains has been obtained by G. James [20]. These DB can also be described as "weakly localised" for small amplitude since although they are exponentially localised the spatial decay rate goes to zero with the amplitude.…”

Section: N +1mentioning

confidence: 87%

“…Next, it is easy to check that these loops form a family of exact periodic solutions of the AL equation only when k = 0 or π, and if the time s is scaled properly as in eq. (20). Moreover, the amplitude of these solutions is exponentially localised in space about position n c = Q, thus these loops constitute a continuous family of discrete breathers parametrised by a translation coordinate Q (see Fig.…”

Section: Salerno's Modelmentioning

confidence: 99%

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Effective Hamiltonian for travelling discrete breathers

MacKay¹,

Sepulchre²

2002

J. Phys. A: Math. Gen.

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Hamiltonian chains of oscillators in general probably do not sustain exact travelling discrete breathers. However solutions which look like moving discrete breathers for some time are not difficult to observe in numerics. In this paper we propose an abstract framework for the description of approximate travelling discrete breathers in Hamiltonian chains of oscillators. The method is based on the construction of an effective Hamiltonian enabling one to describe the dynamics of the translation degree of freedom of moving breathers. Error estimate on the approximate dynamics is also studied. The concept of Peierls-Nabarro barrier can be made clear in this framework. We illustrate the method on two simple examples, namely the Salerno model which interpolates between the Ablowitz-Ladik lattice and the discrete nonlinear Schrödinger system, and the Fermi-Pasta-Ulam chain.

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Section: N +1mentioning

confidence: 87%

Section: Salerno's Modelmentioning

confidence: 99%

Effective Hamiltonian for travelling discrete breathers

MacKay¹,

Sepulchre²

2002

J. Phys. A: Math. Gen.

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“…In the nonlinear case (6.11), one can locally prove a similar result where the center space is replaced by a two-dimensional invariant center manifold [29]. For μ ≈ 0, there exists a smooth local manifold M μ ⊂ D (which can be written as a graph over X c ) invariant under F ω and the symmetries R, T .…”

Section: Example In Infinite Dimensionsmentioning

confidence: 85%

“…Their first components satisfy u 3 −n+1 = −u 3 n and u 4 −n = −u 4 n . We sum up these results in the following lemma [28,29]. Note that for B < 0 and μ > 0 (μ ≈ 0), the local stable and unstable manifolds of (a n , b n ) = 0 do not intersect and thus (6.15) has no small amplitude homoclinic solution.…”

Section: Breathers and "Dark" Breathers In Fermi-pasta-ulam Lattices mentioning

confidence: 90%

“…The reduction scheme described above allows one to prove for the FPU model (with hardening interaction potentials) the existence of small amplitude "breathers" bifurcating above the top of the phonon frequency band [29] (other types of proofs based on variational methods are also available [30,31]). A breather consists of a time-periodic solution of (6.1) (excluding equilibria) which is spatially localized, i.e.…”

Section: Introductionmentioning

confidence: 99%

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Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

James

Sire

The Fermi-Pasta-Ulam Problem

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Abstract. Center manifold theory has been used in recent works to analyze small amplitude waves of different types in nonlinear (Hamiltonian) oscillator chains. This led to several existence results concerning traveling waves described by scalar advance-delay differential equations, pulsating traveling waves determined by systems of advance-delay differential equations, and time-periodic oscillations (including breathers) obtained as orbits of iterated maps in spaces of periodic functions. The Hamiltonian structure of the governing equations is not taken into account in the analysis, which heavily relies on the reversibility of the system. The present work aims at giving a pedagogical review on these topics. On the one hand, we give an overview of existing center manifold theorems for reversible infinite-dimensional differential equations and maps. We illustrate the theory on two different problems, namely the existence of breathers in Fermi-Pasta-Ulam lattices and the existence of traveling breathers (superposed on a small oscillatory tail) in semi-discrete KleinGordon equations. IntroductionThe seminal work of Fermi, Pasta and Ulam in 1955 [12] had a broad impact on the development of the theory of nonlinear lattices. The Fermi-PastaUlam (FPU) model consists of a chain of masses nonlinearly coupled to their nearest neighbors. For a general nearest-neighbors interaction potential V , the governing equations readwhere we note x n the displacement of the nth mass with respect to an equilibrium position (in this version of the model the chain is of infinite extent). The anharmonic interaction potential V satisfies V (0) = 0, V (0) = 1. System (6.1) is Hamiltonian, with total energy

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Macroscopic Behavior of Microscopic Oscillations in Harmonic Lattices via Wigner-Husimi Transforms

Mielke

2006

Arch Rational Mech Anal

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Existence of breathers on FPU lattices

Cited by 56 publications

References 13 publications

Effective Hamiltonian for travelling discrete breathers

Effective Hamiltonian for travelling discrete breathers

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

Macroscopic Behavior of Microscopic Oscillations in Harmonic Lattices via Wigner-Husimi Transforms

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