Solitary Waves in the Nonlinear Dirac Equation

Cuevas–Maraver, Jesús; Boussaïd, Nabile; Comech, Andrew; Lan, Ruomeng; Kevrekidis, P. G.; Saxena, Avadh

doi:10.1007/978-3-319-66766-9_4

Cited by 17 publications

(19 citation statements)

References 170 publications

(305 reference statements)

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“…It was shown that there exist regions where gap solitons and defect modes are stable, and that the degree of nonlocality affects these regions drastically. The gap solitons have also been considered in the framework of PT-symmetric nonlinear Dirac model in [119].…”

Section: Generalized Nonlinear Potentialsmentioning

confidence: 99%

Nonlinear switching and solitons in PT‐symmetric photonic systems

Suchkov

Sukhorukov

Huang

et al. 2016

Laser & Photonics Reviews

352

263

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One of the challenges of the modern photonics is to develop all-optical devices enabling increased speed and energy efficiency for transmitting and processing information on an optical chip. It is believed that the recently suggested ParityTime (PT) symmetric photonic systems with alternating regions of gain and loss can bring novel functionalities. In such systems, losses are as important as gain and, depending on the structural parameters, gain compensates losses. Generally, PT systems demonstrate nontrivial non-conservative wave interactions and phase transitions, which can be employed for signal filtering and switching, opening new prospects for active control of light. In this review, we discuss a broad range of problems involving nonlinear PT-symmetric photonic systems with an intensitydependent refractive index. Nonlinearity in such PT symmetric systems provides a basis for many effects such as the formation of localized modes, nonlinearly-induced PT-symmetry breaking, and all-optical switching. Nonlinear PT-symmetric systems can serve as powerful building blocks for the development of novel photonic devices targeting an active light control.

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Section: Generalized Nonlinear Potentialsmentioning

confidence: 99%

Nonlinear switching and solitons in PT‐symmetric photonic systems

Suchkov

Sukhorukov

Huang

et al. 2016

Laser & Photonics Reviews

352

263

View full text Add to dashboard Cite

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“…The half-soliton considered here is especially interesting as at the NLS level it turns out to be spectrally stable, yet nonlinearly unstable. This issue of the (more involved at the NLDE level [55]) spectral stability of these stationary states is also an especially interesting direction for future study.…”

Section: Conclusion and Future Challengesmentioning

confidence: 99%

Dynamics of Dirac solitons in networks

Sabirov¹,

Babajanov²,

Matrasulov³

et al. 2018

J. Phys. A: Math. Theor.

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We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Stationary soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. The constraints for bond nonlinearity coefficients, conjectured to represent necessary conditions for allowing reflectionless transmission over a Y-junction are derived. The Y-junction considerations are also generalized to a tree network. The analytical results are confirmed by direct numerical simulations. PACS numbers:

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“…(2) The U(1) symmetry and the translation symmetry of the model result in zero eigenvalues with q = 0 and |q| = 1, respectively (in both S = 0 and S = 1 cases). (3) As in the 1d NLD equation, there are also the eigenvalues λ = ±2ωi which are associated with the SU(1, 1) symmetry of the model [47]. This eigenvalue pair corresponds to q = −(2S + 1), i.e., to a highly excited linearization eigenstate.…”

mentioning

confidence: 99%

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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Solitary Waves in the Nonlinear Dirac Equation

Cited by 17 publications

References 170 publications

Nonlinear switching and solitons in PT‐symmetric photonic systems

Nonlinear switching and solitons in PT‐symmetric photonic systems

Dynamics of Dirac solitons in networks

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

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