On the nonexistence of degenerate phase-shift discrete solitons in a dNLS nonlocal lattice
Introduction
The discrete nonlinear Schrödinger (DNLS) is a prototypical nonlinear lattice dynamical model whose analytical and numerical tractability has enabled a considerable amount of progress towards understanding lattice solitary waves/coherent structures [1]. Its apparent simplicity in incorporating the interplay of nonlinearity and discrete dispersion, together with its relevance as an approximation of optical waveguide systems [[2], [3], [4]] and atomic systems in optical lattices [5] have significantly contributed to the popularity of the model. Moreover, its ability to capture numerous linear and nonlinear experimentally observed features has made it a useful playground for a diverse host of phenomena. Such examples include, but are not limited to discrete diffraction [6] and its management [7], discrete solitons [[6], [8], [9]] and vortices [[10], [11]], Talbot revivals [12], and -symmetry and its breaking [13], among many others.
Among the solutions that are of particular interest within the dNLS model are the so-called phase-shift ones, for which the solution does not bear a simply real profile (with phases and , or sites that are in- and out-of phase), but rather is genuinely complex featuring nontrivial phases [1]. Such solutions are more widely known as “discrete vortices” due to the fact that in order to ensure single-valuedness of the solution, upon rotation over a closed discrete contour, they involve variation of the phase by a multiple of . Such waveforms were originally proposed theoretically in [[14], [15]] and subsequently observed experimentally, especially in the setting of optically induced photorefractive crystals [[10], [16]]. A systematic analysis of their potential existence in the standard nearest-neighbor dNLS model was provided in [17] and the relevant results, not only for 2d square lattices, but also for lattices of different types (triangular, honeycomb, etc.) were subsequently summarized in [1]. One of the main findings in this context is that the (symmetric) square vortices (with phase shifts between adjacent excited nodes) indeed persist at high order in case of the square lattice for different topological charges. Another relevant conclusion was that hexagonal and honeycomb lattices present the potential for vortices of topological charge both and .Intriguingly, among the two and for focusing nonlinearity, the latter was found to be more stable than the former (whereas the stability conclusions were reversed in the case of self-defocusing nonlinearity.
However, the relevant analysis poses some intriguing mathematical questions. In particular, the consideration of the most canonical -site vortex in the 2d nearest-neighbor dNLS reveals (due to the relative phase change between adjacent sites of ) a degeneracy of the relevant waveforms and of their potential persistence. By degeneracy, here we mean that the standard approach of looking for critical points of the averaged (along the unperturbed periodic solution) perturbation (see [[18], [19]]), produces one-parameter families of solutions, where the Implicit Function Theorem cannot be applied. The tangent direction to the family represents a direction of degeneracy. Typically this calculation proceeds from the so-called anti-continuum limit [20] in powers of the coupling. Given then the degeneracy of these vortex states, a lingering question is whether such states will persist to all orders or whether they may be destroyed (i.e., the relevant persistence conditions will not be satisfied) at a sufficiently high order. While to all the leading orders considered in [[1], [17]] these solutions persist, in case of high degeneracy expansions up to high orders ( in the -vortex, being the perturbation parameter) are necessary to reach a definitive answer to this question.
Inspired by the inherent difficulty of tackling the 2d problem, here we will opt to examine a simpler 1d problem. As a “caricature” of the 2d interaction, where the fourth site of a given square contour couples back into the first site, we choose to examine a one-dimensional model involving interactions not only of nearest neighbors (NN), but also of neighbors that are next-to-next-nearest (NNNN) ones. In principle, isolating a quadruplet of sites, we reconstruct a geometry similar to the 2d contour. The question that we ask in this simpler (per its 1d nature) setting is whether phase-shift solutions will exist. Surprisingly, and differently from the 2d scenario where at least the vortex was shown to exist, the answer that we find here is always in the negative. Using both a more direct, yet more restrictive, method involving a conserved flux quantity, as well as a more elaborate, yet less restrictive technique based on Lyapunov–Schmidt reductions, we illustrate that such vortical states are always precluded from existence at a sufficiently high order. Since the continuation problem requires to explore the persistence of the solution at high orders in , where the differences with respect to the 2d model, in terms of lattice shape and interaction among sites, play a role, it is perhaps not surprising that we report here a different result in comparison to the -vortex solution of the 2d lattice.
Our presentation will be structured as follows. In Section 2, we will discuss the model and the principal result. In Section 3, we will provide a perturbative approach which, combined with the conservation of the density current, leads towards the formulation of a finite regularity () version of this result. In Section 4, we will overcome the technical limitation of the above regularity requirement by extending considerations to Lyapunov–Schmidt decompositions. Finally, in Section 5, we will summarize our findings and present our conclusions, as well as some emerging questions for future work.
Section snippets
Theoretical setup and principal results
As explained in the previous section, the aim of the work is to investigate the existence of discrete solitons in the NN and NNNN dNLS model of the form: with vanishing boundary conditions at infinity , in the anti-continuum limit, namely . The equations can be written in the Hamiltonian form with
The original motivation leading to the study of the above
A perturbative approach: finite regularity result
In the present Section we tackle the problem with a perturbative approach. For this purpose, we assume to deal with a continuation of which is at least in and write The continuation is assumed to be performed at fixed period (frequency). The results (that are weaker than Theorem 2.1) are collected in the following
Theorem 3.1 For small enough ( ), the only unperturbed solutions (6) that can be continued at fixed period to solutions of
Proof of Theorem 2.1 via Lyapunov–Schmidt decomposition
In the present sectionwe give the proof of Theorem 2.1, without the regularity restriction of the previous section. Consider again the stationary equation (5), now written as and . Since we are interested in the continuation of vortex-like solutions from the anti-continuum limit, we introduce the translation where is a fixed solution of the unperturbed equation, as in formula (6), and represents a small displacement around it, namely
Discussion and conclusions
In the present work, we have revisited the topic of discrete solitons and vortices in lattice models of the discrete nonlinear Schrödinger type. Motivated by the interest in examining 2d asymmetric and super-symmetric configurations, but also by the desire to have a setting that is more analytically tractable, we came up with a 1d toy model. The latter emulates a key feature of the 2d lattice through the inclusion (in a 1d chain) of interactions with the next-to-next-nearest neighbor. To
Acknowledgments
We thank Prof. Dmitry Pelinovsky for the helpful discussions on the proof of Lemma 4.4 that we had during the SIAM conference on Nonlinear Waves and Coherent Structures. This work has been partially supported by NSF- PHY-1602994 (PGK). Also, VK and PGK acknowledge that this work made possible by NPRP grant # [9-329-1-067] from Qatar National Research Fund (a member of Qatar Foundation).
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2020, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :The normal form at order two, as we see from the previous equations, introduces the dependence on q3 that was missing at order one. With standard arguments of bifurcation theory (the same already used for example in [35,36]) it is possible to prove that for ε small enough all families break down and only solutions with qj ∈ {0, π} survive; continuation then follows by means of Newton-Kantorovich fixed point method, since suitable spectral conditions are verified. We refer the reader to Section 5 for details and for interesting results about the role of γ in the linear stability analysis of the solutions: indeed we will show that neither changing the sign in the nonlinear parameter γ (from focusing to defocusing) nor in the coupling parameter ε (from attractive to repulsive) affects the nature of the degenerate eigenspace related to q3, while nondegenerate directions (as already known from the literature, see [24,31]) switch from saddle to center depending on the sign of the product γε.
On the nonexistence of degenerate phase-shift multibreathers in Klein–Gordon models with interactions beyond nearest neighbors
2019, Physica D: Nonlinear PhenomenaCitation Excerpt :The present paper represents a natural follow up of [43], where we studied the related problem of the nonexistence of degenerate phase-shift discrete solitons in a beyond-nearest-neighbor dNLS lattice. We recall that in [43] the nonexistence of phase-shift discrete solitons, which was not easily achievable by means of averaging methods due to the degeneracy of the problem, was obtained in an efficient way by exploiting the rotational symmetry of the model and the density current conservation along the spatial profile of any candidate soliton. The absence of these ingredients in Klein–Gordon models represents an additional layer of difficulty to the degeneracy that one has to face in the continuation problem that we here address.
On the continuation of degenerate periodic orbits via normal form: full dimensional resonant tori
2018, Communications in Nonlinear Science and Numerical SimulationCitation Excerpt :Hence, hypothesis (12) can be verified numerically, by tracking the dependence of the approximate Floquet spectrum on ε in a neighbourhood of the origin. The normal form algorithm here developed, if suitably extended to completely resonant low-dimensional tori, could also allow to deal with degenerate scenarios which emerge studying discrete solitons in 1D nonlocal discrete nonlinear Schroedinger lattices (see [33]) and Multibreathers in 1D Klein-Gordon lattices (like zigzag KG, see [32]): in these models, one parameter families of solutions of the averaged Hamiltonian appear when in the model long range interactions (like next-to-nearest neighbourhood) are added. More naturally, one parameter families of approximate solutions, like the ones observed in the application developed in Section 4, appear in the investigation of vortexes in 2D square lattices [30].
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