Dirac solitons in square binary waveguide lattices

Tran, Truong X.; Nguyen, Xuan N.; Biancalana, Fabio

doi:10.1103/physreva.91.023814

Cited by 22 publications

(9 citation statements)

References 36 publications

(47 reference statements)

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“…The fundamental difference of those models is that they correspond to massless equations, contrary to the Soler model. For this reason, we have confirmed our stability conclusions by comparison with those emerging from the model for square binary waveguides [40], which leads to a massive nonlinear Dirac equation with the same nonlinearity as in Refs. [25,26].…”

supporting

confidence: 86%

“…In that connection, we highlight that although our models of choice may bear a particular nonlinearity, our results suggest that under different nonlinearities including the more physically relevant ones of, e.g., Refs. [25,26] and massive models [40], they still bear stable solitary waves for a suitable wide parametric range of frequencies. Thus, the conclusion of higher-dimensional stability is more general than the specifics of our particular nonlinearity and hence of broad interest.…”

mentioning

confidence: 99%

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Stability of Solitary Waves and Vortices in a 2D Nonlinear Dirac Model

et al. 2016

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We explore a prototypical two-dimensional massive model of the nonlinear Dirac type and examine its solitary wave and vortex solutions. In addition to identifying the stationary states, we provide a systematic spectral stability analysis, illustrating the potential of spinor solutions to be neutrally stable in a wide parametric interval of frequencies. Solutions of higher vorticity are generically unstable and split into lower charge vortices in a way that preserves the total vorticity. These conclusions are found not to be restricted to the case of cubic two-dimensional nonlinearities but are found to be extended to the case of quintic nonlinearity, as well as to that of three spatial dimensions. Our results also reveal nontrivial differences with respect to the better understood nonrelativistic analogue of the model, namely the nonlinear Schrödinger equation. DOI: 10.1103/PhysRevLett.116.214101 Introduction.-In the context of dispersive nonlinear wave equations, admittedly the prototypical model that has attracted a wide range of attention in optics, atomic physics, fluid mechanics, condensed matter, and mathematical physics is the nonlinear Schrödinger equation (NLS) [1][2][3][4][5][6][7]. By comparison, far less attention has been paid to its relativistic analogue, the nonlinear Dirac equation (NLD) [8], despite its presence for almost 80 years in the context of high-energy physics [9][10][11][12][13]. This trend is slowly starting to change, arguably, for three principal reasons. Firstly, significant steps have been taken in the nonlinear analysis of stability of such models [14][15][16][17][18][19], especially in the one-dimensional setting. Secondly, computational advances have enabled a better understanding of the associated solutions and their dynamics [20][21][22][23][24]. Thirdly, and perhaps most importantly, NLD starts emerging in physical systems that arise in a diverse set of contexts of considerable interest. These contexts include, in particular, bosonic evolution in honeycomb lattices [25,26] and a growing class of atomically thin 2D Dirac materials [27], such as graphene, silicene, germanene, and transition metal dichalcogenides [28] (notice that in this Letter, we use nD to refer to n spatial dimensions). Recently, the physical aspects of nonlinear optics, such as light propagation in honeycomb photorefractive lattices (the socalled photonic graphene) [29,30], have prompted the consideration of intriguing dynamical features, e.g., conical diffraction in 2D honeycomb lattices [31]. Inclusion of nonlinearity is then quite natural in these models, although in a number of them (e.g., in atomic and optical physics) the nonlinearity does not couple the spinor components.

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confidence: 86%

mentioning

confidence: 99%

Stability of Solitary Waves and Vortices in a 2D Nonlinear Dirac Model

et al. 2016

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“…• Im(v n,m ) = 0 if n is even and m is odd, resembling the properties of the soliton of Fig. 2 of [18] in the large C limit. This also endows the solitons' real and imaginary parts with a "staggered" structure with alternating rows missing, as seen in the middle panels of Fig.…”

Section: B Discrete Modelmentioning

confidence: 60%

“…Another interesting structure is the 2-site soliton, which, in the continuum limit is reminiscent of the soliton of Fig. 5 in [18]. In the AC limit, such a wave structure is given by u 0,0 = u 0,1 = √ 1 − ω, while once again the v field is vanishing.…”

Section: B Discrete Modelmentioning

confidence: 94%

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Solitary waves in a two-dimensional nonlinear Dirac equation: from discrete to continuum

Cuevas–Maraver¹,

Aceves²,

Saxena³

2017

J. Phys. A: Math. Theor.

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In the present work, we explore a nonlinear Dirac equation motivated as the continuum limit of a binary waveguide array model. We approach the problem both from a near-continuum perspective as well as from a highly discrete one. Starting from the former, we see that the continuum Dirac solitons can be continued for all values of the discretization (coupling) parameter, down to the uncoupled (so-called anti-continuum) limit where they result in a 9-site configuration. We also consider configurations with 1-or 2-sites at the anti-continuum limit and continue them to large couplings, finding that they also persist. For all the obtained solutions, we examine not only the existence, but also the spectral stability through a linearization analysis and finally consider prototypical examples of the dynamics for a selected number of cases for which the solutions are found to be unstable.

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