scholarly journals Symmetric solutions for a 2D critical Dirac equation

2021 ◽  
pp. 2150019
Author(s):  
William Borrelli
Keyword(s):  
Dirac Equation ◽  
Variational Methods ◽  
Exponential Decay ◽  
Two Dimensions ◽  
Effective Model ◽  
Sobolev Embedding ◽  
Honeycomb Structures

In this paper, we show the existence of infinitely many symmetric solutions for a cubic Dirac equation in two dimensions, which appears as effective model in systems related to honeycomb structures. Such equation is critical for the Sobolev embedding and solutions are found by variational methods. Moreover, we also prove smoothness and exponential decay at infinity.

WELL-POSEDNESS OF THE CAUCHY PROBLEM FOR THE CHERN–SIMONS–DIRAC SYSTEM IN TWO DIMENSIONS

2013 ◽  
Vol 10 (04) ◽  
pp. 735-771 ◽  
Author(s):  
MAMORU OKAMOTO
Keyword(s):  
Cauchy Problem ◽  
Dirac Equation ◽  
Sobolev Space ◽  
Gauge Invariance ◽  
Two Dimensions ◽  
Well Posedness ◽  
Dirac System ◽  
The Cauchy Problem ◽  
Chern Simons ◽  
Well Posed

We consider the Cauchy problem associated with the Chern–Simons–Dirac system in ℝ1+2. Using gauge invariance, we reduce the Chern–Simons–Dirac system to a Dirac equation and we uncover the null structure of this Dirac equation. Next, relying on null structure estimates, we establish that the Cauchy problem associated with this Dirac equation is locally-in-time well-posed in the Sobolev space Hs for all s > 1/4. Our proof uses modified L4-type estimates.

scholarly journals Jacob’s Ladder: Prime Numbers in 2D

2020 ◽  
Vol 25 (1) ◽  
pp. 5
Author(s):  
Alberto Fraile ◽  
Roberto Martínez ◽  
Daniel Fernández
Keyword(s):  
Computer Simulations ◽  
Exponential Decay ◽  
Prime Numbers ◽  
Two Dimensions ◽  
Gap Size ◽  
Property A ◽  
Simple Representation ◽  
Jacob's Ladder

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense variety of problems. In this work, we present a simple representation of prime numbers in two dimensions that allows us to formulate a number of conjectures that may lead to important avenues in the field of research on prime numbers. In particular, although the zeroes in our representation grow in a somewhat erratic, hardly predictable way, the gaps between them present a remarkable and intriguing property: a clear exponential decay in the frequency of gaps vs. gap size. The smaller the gaps, the more frequently they appear. Additionally, the sequence of zeroes, despite being non-consecutive numbers, contains a number of primes approximately equal to n / log n , n being the number of terms in the sequence.

scholarly journals A semi-phenomenological approach to explain the event-size distribution of the Drossel-Schwabl forest-fire model

2011 ◽  
Vol 18 (3) ◽  
pp. 381-388 ◽  
Author(s):  
S. Hergarten ◽  
R. Krenn
Keyword(s):  
Size Distribution ◽  
Forest Fire ◽  
Exponential Decay ◽  
Phenomenological Approach ◽  
Two Dimensions ◽  
Scaling Exponent ◽  
Spatial Correlations ◽  
Perfect Agreement ◽  
Fire Model ◽  
Forest Fire Model

Abstract. We present a novel approach to explain the complex scaling behavior of the Drossel-Schwabl forest-fire model in two dimensions. Clusters of trees are characterized by their size and perimeter only, whereas spatial correlations are neglected. Coalescence of clusters is restricted to clusters of similar sizes. Our approach derives the value of the scaling exponent τ of the event size distribution directly from the scaling of the accessible perimeter of percolation clusters. We obtain τ = 1.19 in the limit of infinite growth rate, in perfect agreement with numerical results. Furthermore, our approach predicts the unusual transition from a power law to an exponential decay even quantitatively, while the exponential decay at large event sizes itself is reproduced only qualitatively.

scholarly journals Effective model for crumpling in two dimensions?

1992 ◽  
Vol 46 (10) ◽  
pp. 4761-4764 ◽  
Author(s):  
C. F. Baillie ◽  
D. A. Johnston
Keyword(s):  
Two Dimensions ◽  
Effective Model

STATIONARY STATES OF NONLINEAR DIRAC EQUATIONS WITH GENERAL POTENTIALS

2008 ◽  
Vol 20 (08) ◽  
pp. 1007-1032 ◽  
Author(s):  
YANHENG DING ◽  
JUNCHENG WEI
Keyword(s):  
Dirac Equation ◽  
Variational Methods ◽  
Stationary States ◽  
Real Matrix ◽  
Dirac Equations ◽  
Nonlinear Dirac Equation ◽  
Matrix Potential ◽  
Nonlinear Dirac Equations

We establish the existence of stationary states for the following nonlinear Dirac equation [Formula: see text] with real matrix potential M(x) and superlinearity g(x,|u|)u both without periodicity assumptions, via variational methods.

Sharp exponential decay for solutions of the stationary perturbed Dirac equation

2020 ◽  
pp. 2050078
Author(s):  
Biagio Cassano
Keyword(s):  
Dirac Equation ◽  
Dirac Operator ◽  
Exponential Decay ◽  
Hermitian Matrix ◽  
Dimension Formula ◽  
Massless Dirac Operator

We determine the largest rate of exponential decay at infinity for non-trivial solutions to the Dirac equation [Formula: see text] being [Formula: see text] the massless Dirac operator in dimension [Formula: see text] and [Formula: see text] a (possibly non-Hermitian) matrix-valued perturbation such that [Formula: see text] at infinity, for [Formula: see text]. Also, we show that our results are sharp for [Formula: see text], providing explicit examples of solutions that have the prescripted decay, in presence of a potential with the related behavior at infinity. As a consequence, we investigate the exponential decay at infinity for the eigenfunctions of the perturbed massive Dirac operator, and determine the sharpest possible decay in the case that [Formula: see text] and [Formula: see text].

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