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The Dirac equation in solid state physics and non-linear optics

Abstract : Recently, new two-dimensional materials possessing unique properties have been discovered, the most famous being the graphene. In this materials, electrons at the Fermi level behave as massless particles and can be described by the massless Dirac equation. This phenomenon is quite general, and it is a common features of "honeycomb" periodic structures. Moreover, taking into account interaction leads to non-linear Dirac equations, which also appear in the description of light propagation in particular waveguides. The aim of the thesis is to study existence and stability of stationary solutions for those equations with both sub-critical and critical nonlinearities, and to show that they are limit of stationary solutions to the Schroedinger equation with honeycomb potential, for a suitable choice of parameters. This amounts to solving the Euler-Lagrange equation for strongly indefinite energy functionals, involving the Dirac operator. We will deal with critical nonlinearities, which are still poorly understood, and appear naturally in non-linear optics. This results may have an impact on the understanding some solid state or nonlinear optics systems.
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Submitted on : Thursday, October 25, 2018 - 3:13:41 PM
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William Borrelli. The Dirac equation in solid state physics and non-linear optics. Analysis of PDEs [math.AP]. Université Paris sciences et lettres, 2018. English. ⟨NNT : 2018PSLED021⟩. ⟨tel-01905035⟩

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