Wave Modulations in Anharmonic Lattices

Tsurui, Akira

doi:10.1143/ptp.48.1196

Cited by 95 publications

(51 citation statements)

References 3 publications

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“…Such a threshold appears in 2D and 3D systems. As it was mentioned in [6], this explanation well corroborates with analogous phenomenon known for discrete systems, where existence of discrete envelope solitons of arbitrarily small amplitudes is possible in the 1D case [10] but in two dimensions requires a threshold intensity [11].…”

Section: Introductionsupporting

confidence: 84%

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Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

Bludov

Brazhnyi

2007

Phys. Rev. A

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It is shown that inhomogeneous nonlinear interactions in a Bose-Einstein condensate loaded in an optical lattice can result in delocalizing transition in one dimension, what sharply contrasts to the known behavior of discrete and periodic systems with homogeneous nonlinearity. The transition can be originated either by decreasing the amplitude of the linear periodic potential or by the change of the mean value of the periodic nonlinearity. The dynamics of the delocalizing transition is studied.

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Section: Introductionsupporting

confidence: 84%

“…For the existence of small amplitude solitons this condition must be satisfied near at least one of the gap edges. This happens, in particular, in models with the homogeneous nonlinearity (G ≡const) [3,9] and in discrete models with on-cite nonlinearity [10]. Thus, the requirements…”

Section: The Modelmentioning

confidence: 99%

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

Bludov

Brazhnyi

2007

Phys. Rev. A

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“…(1) 0 is purely due to cubic interaction potential nonlinearity (and disappears for p 0 = 0). This is known in solid state physics [11,12].…”

Section: Multiple Scale Expansion -Derivation Of a Nonlinear Schmentioning

confidence: 99%

“…Dust-lattice waves (DLWs) are reminiscent of waves ('phonons') propagating in atomic chains, which are long known to be dominated by interesting nonlinear phenomena (localized modes, instabilities), due to the intrinsic nonlinearities of inter-atomic interaction mechanisms and/or on-site substrate potentials [11,12,13,14]. The present study is devoted to the study of one such phenomenon, namely the nonlinear amplitude modulation of weakly nonlinear oscillations with respect to longitudinal lattice oscillations.…”

Section: Introductionmentioning

confidence: 99%

Modulated wavepackets associated with longitudinal dust grain oscillations in a dusty plasma crystal

Kourakis

Shukła

2004

Physics of Plasmas

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The nonlinear amplitude modulation of longitudinal dust lattice waves (LDLWs) propagating in a dusty plasma crystal is investigated in a continuum approximation. It is shown that long wavelength LDLWs are modulationally stable, while shorter wavelengths may be unstable. The possibility for the formation and propagation of different envelope localized excitations is discussed. It is shown that the total grain displacement bears a (weak) constant displacement (zeroth harmonic mode), due to the asymmetric form of the nonlinear interaction potential. The existence of asymmetric envelope localized modes is predicted. The types and characteristics of these coherent nonlinear structures are discussed.

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“…The condition B > 0 corresponds to the modulational instability of the nonlinear normal modes of frequency ω ≈ ω 0 , yielding their spatial localization (see [50] for a formal study through multi-scale expansions). This condition leads to a similar instability for binary oscillations [51].…”

Section: Breathers and "Dark" Breathers Corresponding To Homoclinicsmentioning

confidence: 99%

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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Wave Modulations in Anharmonic Lattices

Cited by 95 publications

References 3 publications

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

Delocalizing transition in one-dimensional condensates in optical lattices due to inhomogeneous interactions

Modulated wavepackets associated with longitudinal dust grain oscillations in a dusty plasma crystal

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

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