Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators

James, Guillaume; Sire, Yannick

doi:10.1007/s00220-004-1260-y

Cited by 28 publications

(85 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Subsequently, the analysis of traveling waves by Iooss and Kirchgässner has been extended to pulsating traveling waves by the present authors [19,20,21]. These solutions are searched to satisfy x n (t) = x n−p (t − T ), (6.4) where p ≥ 1 is a fixed integer and the propagation time T ∈ R * is taken as a parameter (traveling waves correspond to fixing p = 1 and c = 1/T in the above definition).…”

Section: Introductionmentioning

confidence: 93%

“…In the following we restrict ourselves to the case p = 2 [20]. Our analysis generalizes the one performed by Iooss and Kirchgässner [18], who treated the case of traveling waves (p = 1).…”

Section: Example In Infinite Dimensionsmentioning

confidence: 98%

“…In the following section, we start by an illustration of center manifold theory on a class of reversible advance-delay differential equations. The material is issued from [18,20,21].…”

Section: Center Manifold Reduction For Infinite-dimensional Differentmentioning

confidence: 99%

“…On the one hand, we present a review of [29] (Sects. 6.2.2 and 6.4.1) and [20] (Sects. 6.3.1 and 6.4.2), based on center manifold theory, which examine the existence of breathers and nanopterons in FPU and KG lattices respectively.…”

Section: Introductionmentioning

confidence: 97%

“…The center manifold theorem reduces systems (6.3)-(6.4) locally to a finite-dimensional ordinary differential equation on a center manifold, whose dimension depends on p and parameters γ, T . In certain parameter regimes, this simplification allows one to show the existence of small amplitude traveling waves [18] and pulsating traveling waves [20] (for an even potential V ) in the form of nanopterons, a denomination introduced by Boyd [22]. Nanopterons consist in a single pulse (here a modulated plane wave whose envelope is spatially localized), superposed on a nondecaying oscillatory tail which is small with respect to the central pulse amplitude.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

James

Sire

The Fermi-Pasta-Ulam Problem

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Introductionmentioning

confidence: 93%

“…In the following we restrict ourselves to the case p = 2 [20]. Our analysis generalizes the one performed by Iooss and Kirchgässner [18], who treated the case of traveling waves (p = 1).…”

Section: Example In Infinite Dimensionsmentioning

confidence: 98%

“…In the following section, we start by an illustration of center manifold theory on a class of reversible advance-delay differential equations. The material is issued from [18,20,21].…”

Section: Center Manifold Reduction For Infinite-dimensional Differentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations