An extensive resonant normal form for an arbitrary large Klein–Gordon model

Paleari, Stefano; Penati, Tiziano

doi:10.1007/s10231-014-0456-9

Cited by 10 publications

(38 citation statements)

References 25 publications

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“…Existence and stability of breathers have been studied in the dKG equation in many recent works. In particular, exploring the limit of weak coupling between the nonlinear oscillators, existence [26] and stability [2,4] of the fundamental (single-site) breathers were established (see also the recent works in [30,31]). More complicated multi-breathers were classified from the point of their spectral stability in the recent works [1,25,33].…”

Section: Introductionmentioning

confidence: 99%

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

2016

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Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are approximated by equations of the discrete nonlinear Schrödinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss applications of the discrete nonlinear Schrödinger equation in the context of existence and stability of breathers of the Klein-Gordon lattice.

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Section: Introductionmentioning

confidence: 99%

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

2016

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“…Thus it is not possible to apply the usual implicit function theorem for their continuation. Nevertheless, it is well known (see [32,35]) that for sufficiently low energies (E ≪ 1) and in a suitable anti-continuum limit regime (namely ǫ ≪ E), (3) can be approximated by (2). Indeed, the dNLS Hamiltonian (2) turns out to be a resonant normal form of (3); this is evident by averaging both the coupling term and the nonlinear term with respect to the periodic flow given by the harmonic part of the Hamiltonian (3).…”

Section: Theoretical Setup and Principal Resultsmentioning

confidence: 99%

On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice

Penati

Sansottera

Paleari

et al. 2018

Physica D: Nonlinear Phenomena

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We consider a one-dimensional discrete nonlinear Schrödinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-sites vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler variant. Due to the specific choice of lengths of the inter-site interactions, the vortex configurations considered present a degeneracy which causes the standard continuation techniques to be non-applicable.In the present one-dimensional case, the existence of a conserved quantity for the soliton profile (the so-called density current), together with a perturbative construction, leads to the nonexistence of any phase-shift discrete soliton which is at least C 2 with respect to the small coupling ǫ, in the limit of vanishing ǫ. If we assume the solution to be only C 0 in the same limit of ǫ, nonexistence is instead proved by studying the bifurcation equation of a Lyapunov-Schmidt reduction, expanded to suitably high orders. Specifically, we produce a nonexistence criterion whose efficiency we reveal in the cases of partial and full degeneracy of approximate solutions obtained via a leading order expansion.

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“…In order to prove such a conjecture, one could still follow this indirect approach by increasing the accuracy of the normal form approximation by adding further non-local linear and nonlinear terms to the dNLS H101, in the spirit of a more general dNLS approximation (see [30,31]). Alternatively, one can use a more direct approach and perform a local normal form technique around the low-dimensional resonant torus, with the advantage of working directly in the original KG model without passing from the dNLS approximation (see [36] for the maximal tori case).…”

Section: Conclusion -Future Directionsmentioning

confidence: 99%

“…The price one has to pay for the use of this strategy lies in the restrictions in the regime of parameters for which the models (1) are well approximated by the corresponding averaged normal forms (7). However, Hamiltonian normal form theory provides the tools needed to modify the approximating models in different regimes of the parameters E and , as in [30,34], thus leading to a new dNLS-type starting model for an indirect approach. Such a new nonlocal dNLS normal form would surely include additional linear and nonlinear corrections with respect to those present in (7).…”

Section: Introductionmentioning

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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An extensive resonant normal form for an arbitrary large Klein–Gordon model

Cited by 10 publications

References 25 publications

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

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

On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice

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

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