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

https://doi.org/10.1016/j.physd.2017.12.012Get rights and content

Highlights

  • A 1-dimensional dNLS model with interactions beyond nearest neighbors is studied.

  • Nonexistence of four-sites phase-shift discrete solitons is proved.

  • A nonexistence criterion, based on the current conservation, is proposed.

  • Lyapunov–Schmidt-based bifurcation analysis confirms the validity of the criterion.

Abstract

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-site 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 C2 with respect to the small coupling ϵ, in the limit of vanishing ϵ. If we assume the solution to be only C0 in the same limit of ϵ, 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.

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 PT-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 0 and π, 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 2π. 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 π2 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 S=1 and S=2.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 4-site vortex in the 2d nearest-neighbor dNLS reveals (due to the relative phase change between adjacent sites of π2) 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 (ϵ6 in the π2-vortex, ϵ 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 π2 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 ϵ, 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 π2-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 (C2) 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: iψ̇j=ψjϵ2(Δ1+Δ3)ψj+34ψj|ψj|2,(Δlψ)jψj+l2ψj+ψjl,with vanishing boundary conditions at infinity ψ2(C), in the anti-continuum limit, namely ϵ0. The equations can be written in the Hamiltonian form iψ̇j=Kψ¯j with K=jZ|ψj|2+ϵ2jZ|ψj+1ψj|2+|ψj+3ψj|2+38jZ|ψj|4.

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 {ϕj(ϵ)}jZ which is at least C2 in ϵ and write ϕj=ϕj(0)+ϵϕj(1)+ϵ2ϕj(2)+o(ϵ2).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 ϵ small enough ( ϵ0), the only unperturbed solutions (6) that can be continued at fixed period to C2 solutions ϕ(ϵ) of

Proof of Theorem 2.1 via Lyapunov–Schmidt decomposition

In the present sectionwe give the proof of Theorem 2.1, without the C2 regularity restriction of the previous section. Consider again the stationary equation (5), now written as ωϕ+ϵLϕ34ϕ|ϕ|2=0,withL12Δ1+Δ3,and ϕ2(C). Since we are interested in the continuation of vortex-like solutions from the anti-continuum limit, we introduce the translation w(ϵ)ϕ(ϵ)v,where v is a fixed solution of the unperturbed equation, as in formula (6), and w(ϵ) represents a small displacement around it, namely vj

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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