Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

Pelinovsky, Dmitry E.; Penati, Tiziano; Paleari, Stefano

doi:10.1142/s0129055x1650015x

Cited by 17 publications

(28 citation statements)

References 42 publications

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“…where ω 0 , k 0 , and m 0 are the same as for (25). P3 There exists a unique eigenvectoru ∈ C ∞ per ((0, T ); 2 (Z)) of the spectral problem (6) with λ = 0, where u ∈ C ∞ per ((0, T ); 2 (Z)) is the breather of the KG lattice equation (1).…”

Section: Asymptotic Expansionsmentioning

confidence: 99%

Nonlinear Instabilities of Multi‐Site Breathers in Klein–Gordon Lattices

2015

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In the present work, we explore the possibility of excited breather states in a nonlinear Klein-Gordon lattice to become nonlinearly unstable, even if they are found to be spectrally stable. The mechanism for this fundamentally nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the KleinGordon lattice with a soft (Morse) and a hard (φ 4 ) potential. Compared to the case of the nonlinear Schrödinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the excited breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, excited breather states are observed to be nonlinearly stable.

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Section: Asymptotic Expansionsmentioning

confidence: 99%

Nonlinear Instabilities of Multi‐Site Breathers in Klein–Gordon Lattices

2015

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“…Despite their widespread use, rigorous justifications of the rotating wave procedures are more sparse, with an early example being [14], wherein Hamiltonian Klein-Gordon lattices are approximated by nonlinear Schrödinger equations (see also [9,17]). A justification for the discrete nonlinear Schrödinger approximation was provided rather recently in [22].…”

Section: Introductionmentioning

confidence: 99%

Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

Muda

Akbar

Kusdiantara

et al. 2020

ASY

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We consider a discrete nonlinear Klein-Gordon equations with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.

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“…using again the letter R to denote the corresponding term of (34). It turns out that in the small energy regime (i.e., for ρ small enough) (36) looks as a ρ 2 -perturbation of the NL-dNLS stationary problem…”

Section: The First Lyapunov-schmidt Decompositionmentioning

confidence: 99%

On the nonexistence of degenerate phase-shift multibreathers in Klein–Gordon models with interactions beyond nearest neighbors

Penati

Koukouloyannis

Sansottera

et al. 2019

Physica D: Nonlinear Phenomena

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In this work, we study the existence of low amplitude four-site phase-shift multibreathers for small values of the coupling in Klein-Gordon (KG) chains with interactions longer than the classical nearest-neighbour ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namely the so-called zigzag lattice, as well as the two-dimensional square lattice. We examine initially the persistence conditions of the system, in order to seek for vortexlike waveforms. Although this approach provides useful insights, due to the degeneracy of these solutions, it does not allow us to determine if they constitute true solutions of our system. In order to overcome this obstacle, we follow a different route. In the case of the zigzag configuration, by means of a Lyapunov-Schmidt decomposition, we are able to establish that the bifurcation equation for our model can be considered, in the small energy and small coupling regime, as a perturbation of a corresponding non-local discrete nonlinear Schrödinger (NL-dNLS) equation. There, nonexistence results of degenerate phase-shift discrete solitons can be demonstrated by exploiting the expansion of a suitable density current of the NL-dNLS, obtained in recent literature. Finally, briefly considering a one-dimensional model bearing similarities to the square lattice, we conclude that the above strategy is not efficient for the proof of the existence or nonexistence of vortices due to the higher degeneracy of this configuration.

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Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

Cited by 17 publications

References 42 publications

Nonlinear Instabilities of Multi‐Site Breathers in Klein–Gordon Lattices

Nonlinear Instabilities of Multi‐Site Breathers in Klein–Gordon Lattices

Reduction of a damped, driven Klein–Gordon equation into a discrete nonlinear Schrödinger equation: Justification and numerical comparison

On the nonexistence of degenerate phase-shift multibreathers in Klein–Gordon models with interactions beyond nearest neighbors

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