Travelling Breathers in Klein-Gordon Lattices as Homoclinic Orbits to p-Tori

Sire, Yannick

doi:10.1007/s10884-005-8268-7

Cited by 11 publications

(25 citation statements)

References 32 publications

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“…The models we consider (26) [FP99,FP02,FP04a,FP04b]), pulsating traveling waves (see [JS05,Sir05]). The relevance of these solutions in biological phenomena has also been discussed (see [DPW92,PS04,Pey04]).…”

Section: (19)mentioning

confidence: 99%

Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

Fontich

Llave

Sire³

2015

Journal of Differential Equations

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Abstract. We construct quasi-periodic and almost periodic solutions for coupled Hamiltonian systems on an infinite lattice which is translation invariant. The couplings can be long range, provided that they decay moderately fast with respect to the distance.For the solutions we construct, most of the sites are moving in a neighborhood of a hyperbolic fixed point, but there are oscillating sites clustered around a sequence of nodes. The amplitude of these oscillations does not need to tend to zero. In particular, the almost periodic solutions do not decay at infinity.The main result is an a-posteriori theorem. We formulate an invariance equation. Solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation. We show that, if there is an approximate solution of the invariance equation that satisfies some nondegeneracy conditions, there is a true solution close by.This does not require that the system is close to integrable, hence it can be used to validate numerical calculations or formal expansions.The proof of this a-posteriori theorem is based on a Nash-Moser iteration, which does not use transformation theory. Simpler versions of the scheme were developed in E. Fontich, R. de la Llave,Y. Sire J. Differential. Equations. 246, 3136 (2009).One technical tool, important for our purposes, is the use of weighted spaces that capture the idea that the maps under consideration are local interactions. Using these weighted spaces, the estimates of iterative steps are similar to those in finite dimensional spaces. In particular, the estimates are independent of the number of nodes that get excited. Using these techniques, given two breathers, we can place them apart and obtain an approximate solution, which leads to a true solution nearby. By repeating the process infinitely often, we can get solutions with infinitely many frequencies which do not tend to zero at infinity.

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Section: (19)mentioning

confidence: 99%

Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

Fontich

Llave

Sire³

2015

Journal of Differential Equations

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“…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: 95%

“…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%

“…For the more technical case p > 2 we refer the reader to [21]. In the case p = 2, solutions can be rewritten [20]:…”

Section: Example In Infinite Dimensionsmentioning

confidence: 99%

“…The case p > 2 is examined in reference [21]. The simplest bifurcation yields a normal form having a similar structure, with p pairs of simple purely imaginary eigenvalues in addition to the bifurcating pair ±iq 0 .…”

Section: Traveling Breathers In Klein-gordon Lattices Local Analyticamentioning

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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Numerical computation of travelling breathers in Klein–Gordon chains

Sire

James

2005

Physica D: Nonlinear Phenomena

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We numerically study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions having the property of being exactly translated by p sites along the chain after a fixed propagation time T (these solutions generalize the concept of solitary waves for which p = 1). In the case of even on-site potentials, the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently (G. James and Y. Sire, to appear in Comm. Math. Phys., 2004), the tail being exponentially small with respect to the central oscillation size. In this paper we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials. We find that Klein-Gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail. We provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations. In addition we numerically observe the existence of these solutions in the case of non even potentials.

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Travelling Breathers in Klein-Gordon Lattices as Homoclinic Orbits to p-Tori

Cited by 11 publications

References 32 publications

Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

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

Numerical computation of travelling breathers in Klein–Gordon chains

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