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

In this paper we prove the existence of an exponentially localized stationary solution for a two-dimensional cubic Dirac equation. It appears as an effective equation in the description of nonlinear waves for some Condensed Matter (Bose-Einstein condensates) and Nonlinear Optics (optical fibers) systems. The nonlinearity is of Kerr-type, that is of the form |$\psi$| 2 $\psi$ and thus not Lorenz-invariant. We solve compactness issues related to the critical Sobolev embedding H 1 2 (R 2 , C 2) $\rightarrow$ L 4 (R 2 , C 4) thanks to a particular radial ansatz. Our proof is then based on elementary dynamical systems arguments. Content

Multiple solutions for a self-consistent Dirac equation in two dimensions

This paper is devoted to the variational study of an effective model for the electron transport in a graphene sample. We prove the existence of infinitely many stationary solutions for a nonlin-ear Dirac equation which appears in the WKB limit for the Schr{\"o}dinger equation describing the semi-classical electron dynamics. The interaction term is given by a mean field, self-consistent potential which is the trace of the 3D Coulomb potential. Despite the nonlinearity being 4-homogeneous, compactness issues related to the limiting Sobolev embedding $H^{\frac{1}{2}}(\Omega,\mathbb{C})\rightarrow L^{4} (\Omega,\mathbb{C})$ are avoided thanks to the regular-ization property of the operator (-\Delta)^{-\frac{1}{2}. This also allows us to prove smoothness of the solutions. Our proof follows by direct arguments

Effect of temperature on non-Markovian dynamics in Coulomb crystals

In this paper we generalize the results reported in Phys. Rev. A 88, 010101 (2013) and investigate the flow of information induced in a Coulomb crystal in presence of thermal noise. For several temperatures we calculate the non-Markovian character of Ramsey interferometry of a single 1/2 spin with the motional degrees of freedom of the whole chain. These results give a more realistic picture of the interplay between temperature, non-Markovianity and criticality.Comment: 5 pages, 3 figures. Accepted for publication in Special Issue of the International Journal of Quantum Information devoted to IQIS2013 conferenc

Some properties of Dirac-Einstein bubbles

We prove smoothness and provide the asymptotic behavior at infinity of solutions of Dirac-Einstein equations on $\mathbb{R}^3$, which appear in the bubbling analysis of conformal Dirac-Einstein equations on spin 3-manifolds. Moreover, we classify ground state solutions, proving that the scalar part is given by Aubin-Talenti functions, while the spinorial part is the conformal image of $-\frac{1}{2}$-Killing spinors on the round sphere $\mathbb{S}^3$.Comment: 14 pages. J. Geom. Anal. (2020

A Novel Method of Solving Linear Programs with an Analog Circuit

We present the design of an analog circuit which solves linear programming (LP) problems. In particular, the steady-state circuit voltages are the components of the LP optimal solution. The paper shows how to construct the circuit and provides a proof of equivalence between the circuit and the LP problem. The proposed method is used to implement a LP-based Model Predictive Controller by using an analog circuit. Simulative and experimental results show the effectiveness of the proposed approach.Comment: 8 page