Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures

Arbunich, Jack; Sparber, Christof

doi:10.1063/1.5021754

Cited by 36 publications

(50 citation statements)

References 29 publications

(57 reference statements)

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“…is frequently used, see for instance [5,23,37]. As shown in Theorem 4.1, we do the error estimates in the microscopic scales with the regular Sobolev spaces.…”

Section: Resultsmentioning

confidence: 99%

“…One of the most successful example is the honeycomb-based material [22,33,38]. It stimulates the mathematical analysis of the Schrödinger equation with a honeycomb potential over the past few years [1,2,5,9,19,20]. Fefferman and Weinstein rigorously proved the existence of Dirac points of Schrödinger operator with a generic honeycomb potential [19], and later gave a mathematical justification of the massless Dirac equation which governs the dynamics of the wave packets associated with Dirac points [20].…”

Section: Introductionmentioning

confidence: 99%

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Wave packet dynamics in slowly modulated photonic graphene

Xie

Zhu

2019

Journal of Differential Equations

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Mathematical analysis on electromagnetic waves in photonic graphene, a photonic topological material which has a honeycomb structure, is one of the most important current research topics. By modulating the honeycomb structure, numerous topological phenomena have been observed recently. The electromagnetic waves in such a media are generally described by the 2-dimensional wave equation. It has been shown that the corresponding elliptic operator with a honeycomb material weight has Dirac points in its dispersion surfaces. In this paper, we study the time evolution of the wave packets spectrally concentrated at such Dirac points in a modulated honeycomb material weight. We prove that such wave packet dynamics is governed by the Dirac equation with a varying mass in a large but finite time. Our analysis provides mathematical insights to those topological phenomena in photonic graphene.

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“…is frequently used, see for instance [5,23,37]. As shown in Theorem 4.1, we do the error estimates in the microscopic scales with the regular Sobolev spaces.…”

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Wave packet dynamics in slowly modulated photonic graphene

Xie

Zhu

2019

Journal of Differential Equations

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“…In that case one may expect to recover the metric graph model in an appropriate scaling regime. We also mention that a rigorous mathematical study of the dynamics and the existence of Dirac solitons on lattices has been recently treated in [12,15,17,16,31].…”

Section: Introductionmentioning

confidence: 99%

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

2019

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We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the subcritical and the critical case, and then on the NLDE, highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit, proving the convergence of bound states.

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“…where Φ j , j = 1, 2 are Bloch functions at the Dirac point (2), and the functions α 0,j are some (complex) amplitudes to be determined. Then one expects the solution to (3) to evolve, at leading order in , as a modulation of Bloch functions:…”

Section: Introductionmentioning

confidence: 99%

“…As explained in [3], performing a multiscale expansion one obtains the following effective Dirac system for the modulation coefficients α j :…”

Section: Introductionmentioning

confidence: 99%

Weakly localized states for nonlinear Dirac equations

Borrelli

2018

Calc. Var.

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We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization as least action critical points of a suitable action functional. We also indicate how the content of the present paper allows to extend our previous results for the massive case [5] to more general nonlinearities.

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Rigorous derivation of nonlinear Dirac equations for wave propagation in honeycomb structures

Cited by 36 publications

References 29 publications

Wave packet dynamics in slowly modulated photonic graphene

Wave packet dynamics in slowly modulated photonic graphene

An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

Weakly localized states for nonlinear Dirac equations

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