Variational approximations in discrete nonlinear Schrödinger equations with next-nearest-neighbor couplings

Chong, Christopher; Carretero-González, R.; Malomed, Boris A.

doi:10.1016/j.physd.2011.04.011

Cited by 21 publications

(24 citation statements)

References 22 publications

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“…The latter setting is very close to genuinely two-dimensional setting in a square lattice plaquette, which constitutes our final example in section VII. We close our presentation by some remarks on the parallels of our results with the simpler case of the discrete nonlinear Schrödinger lattices [17] (section VIII), which has been examined earlier in [7,18], as well as a summary of our conclusions and some future directions (section IX). The Hamiltonian of a Klein-Gordon chain with nearest neighbor interactions is the following…”

Section: Introductionmentioning

confidence: 75%

Multibreathers in Klein–Gordon chains with interactions beyond nearest neighbors

Koukouloyannis

Cuevas

Rothos

2013

Physica D: Nonlinear Phenomena

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We study the existence and stability of multibreathers in Klein-Gordon chains with interactions that are not restricted to nearest neighbors. We provide a general framework where such long range effects can be taken into consideration for arbitrarily varying (as a function of the node distance) linear couplings between arbitrary sets of neighbors in the chain. By examining special case examples such as three-site breathers with next-nearest-neighbors, we find crucial modifications to the nearest-neighbor picture of one-dimensional oscillators being excited either in-or anti-phase. Configurations with nontrivial phase profiles emerge from or collide with the ones with standard (0 or π) phase difference profiles, through supercritical or subcritical bifurcations respectively. Similar bifurcations emerge when examining four-site breathers with either next-nearest-neighbor or even interactions with the three-nearest one-dimensional neighbors. The latter setting can be thought of as a prototype for the two-dimensional building block, namely a square of lattice nodes, which is also examined. Our analytical predictions are found to be in very good agreement with numerical results.

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Section: Introductionmentioning

confidence: 75%

Multibreathers in Klein–Gordon chains with interactions beyond nearest neighbors

Koukouloyannis

Cuevas

Rothos

2013

Physica D: Nonlinear Phenomena

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“…Long-range interactions can have a significant effect on nonlinear excitations and yield phenomena that are rather different from those that result from only nearest-neighbor coupling. For example, stationary solitary waves with a nontrivial phase can arise both in discrete nonlinear Schrödinger (DNLS) equations with nextnearest-neighbor (NNN) interactions [16,30] and in NNN discrete Klein-Gordon (KG) [31] equations, and bistability of solitary waves is possible in DNLS equations with long-range interactions [32,33]. Finally, and most relevant for the present paper, breathers in KG and Fermi-Pasta-Ulam-Tsingou (FPUT) lattices with long-range interactions can exhibit a crossover from exponential decay (at short distances from the breather center) to algebraic decay (at long distances) if the interactions decay significantly slowly (specifically, algebraically slowly) [24].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear excitations in magnetic lattices with long-range interactions

et al. 2019

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We study-experimentally, theoretically, and numerically-nonlinear excitations in lattices of magnets with long-range interactions. We examine breather solutions, which are spatially localized and periodic in time, in a chain with algebraically-decaying interactions. It was established two decades ago (Flach 1998 Phys. Rev. E 58 R4116) that lattices with long-range interactions can have breather solutions in which the spatial decay of the tails has a crossover from exponential to algebraic decay. In this article, we revisit this problem in the setting of a chain of repelling magnets with a mass defect and verify, both numerically and experimentally, the existence of breathers with such a crossover.

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“…The compatibility condition (55) is equivalent to the variational energy method introduced in [18] (see also Remark 2.1). Using the notation of the phase shifts ϕ (see (7)), one easily gets as candidate bifurcation points the same ones shown in Section 3.1, i.e. the two isolated points ϕ ∈ {(0, 0, 0), (π, π, π)}, the three families (18), and their intersections ϕ ∈ {±( π 2 , π 2 , π 2 )} that we call symmetric vortices.…”

Section: Kernel Equationmentioning

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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Variational approximations in discrete nonlinear Schrödinger equations with next-nearest-neighbor couplings

Cited by 21 publications

References 22 publications

Multibreathers in Klein–Gordon chains with interactions beyond nearest neighbors

Multibreathers in Klein–Gordon chains with interactions beyond nearest neighbors

Nonlinear excitations in magnetic lattices with long-range interactions

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

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