Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity

“…More precisely, after a suitable rescaling we exploit the properties of the explicit solution to the (massless) limiting problem. The analysis carried out in the present paper allows us to deal with the general case: 0 < β 2 β 1 , The theorem can be proved studying (113) thanks to the shooting argument given in [5]. The delicate part consists in controlling the error committed in approximating the solution with that of the limiting problem (19) on a suitable interval.…”

Section: The Massive Casementioning

confidence: 97%

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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“…Precisely, [50] suggests the study again of the stationary solutions, that is χ(t, x) = e −iωt ψ(x), with ω ∈ R, that solve Dψ − |ψ| p−2 ψ = ωψ . (5) recall that the existence of stationary solutions for cubic and Hartree-type Dirac equations for honeycomb structures and graphene samples has been investigated in [14,13,15]; whereas, for an overview on global existence results for one dimensional NLDE we refer to [18,44].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

Borrelli¹,

Carlone²,

Tentarelli³

2019

SIAM J. Math. Anal.

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In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the L 2 -subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit. 1The attention recently attracted by the linear and the nonlinear Dirac equations is due to their applications, as effective equations, in many physical models, as in solid state physics and nonlinear optics [33,34].While originally the NLDE appeared as a field equation for relativistic interacting fermions [38], then it was used in particle physics to simulate features of quark confinement, acoustic physics, and in the context of Bose-Einstein condensates [34].Recently, it also appeared that some properties of physical models, as thin carbon structures, are well described using as an effective equation for non-relativistic electronic properties , the Dirac equation. We mention, thereupon, the seminal papers by Fefferman and Weinstein [28,29], the work of Arbunich and Sparber [10] (where a rigorous justification of linear and nonlinear equations in two-dimensional honeycomb structures is given) and the referenced therein. In addition, we

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Stationary solutions for the 2D critical Dirac equation with Kerr nonlinearity

Cited by 23 publications

References 53 publications

Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks

Symmetries of Nonlinear PDEs on Metric Graphs and Branched Networks

Weakly localized states for nonlinear Dirac equations

Nonlinear Dirac Equation on Graphs with Localized Nonlinearities: Bound States and Nonrelativistic Limit

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