10,484 research outputs found
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 || 2 and thus
not Lorenz-invariant. We solve compactness issues related to the critical
Sobolev embedding H 1 2 (R 2 , C 2) 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 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 , 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 -Killing spinors on the round sphere .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
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