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In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on \\mathbb{R}^{1+d} (d\\geq 4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.

The main purpose of the paper Li et al (2020 New J. Phys. 22 113019) is to determine the common properties of electronic waves and electromagnetic waves. The electronic Maxwell’s equations are identical to the Dirac equation, describing both electrons and positrons.

This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either H 1 -like or L 2 and L 2 -like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.

The concept of isogeometric analysis (IGA) was first applied to the approximation of Maxwell equations in [A. Buffa, G. Sangalli, and R. Vázquez, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 1143–1152]. The method is based on the construction of suitable B-spline spaces such that they verify a De Rham diagram. Its main advantages are that the geometry is described exactly with few elements, and the computed solutions are smoother than those provided by finite elements. In this paper we develop the theoretical background to the approximation of vector fields in IGA. The key point of our analysis is the definition of suitable projectors that render the diagram commutative. The theory is then applied to the numerical approximation of Maxwell source problems and eigenproblems, and numerical results showing the good behavior of the scheme are also presented.

We are interested in the existence of normalized solutions to the problem ( - Δ ) m u + μ | y | 2 m u + λ u = g ( u ) , x = ( y , z ) ∈ R K × R N - K , ∫ R N | u | 2 d x = ρ > 0 , in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the L 2 -ball. Moreover, we find also a solution to the related curl–curl problem ∇ × ∇ × U + λ U = f ( U ) , x ∈ R N , ∫ R N | U | 2 d x = ρ , which arises from the system of Maxwell equations and is of great importance in nonlinear optics.

Maxwell’s equations and the Dirac equation are the first-order differential relativistic wave equation for electromagnetic waves and electronic waves respectively. Hence, there is a notable similarity between these two wave equations, which has been widely researched since the Dirac equation was proposed. In this paper, we show that the Maxwell equations can be written in an exact form of the Dirac equation by representing the four Dirac operators with 8 × 8 matrices. Unlike the ordinary 4 × 4 Dirac equation, both spin–1/2 and spin–1 operators can be derived from the 8 × 8 Dirac equation, manifesting that the 8 × 8 Dirac equation is able to describe both electrons and photons. As a result of the restrictions that the electromagnetic wave is a transverse wave, the photon is a spin–1 particle. The four–current in the Maxwell equations and the mass in the electronic Dirac equation also force the electromagnetic field to transform differently to the electronic field. We use this 8 × 8 representation to find that the Zitterbewegung of the photon is actually the oscillatory part of the Poynting vector, often neglected upon time averaging.

Advances in photonics and nanotechnology have the potential to revolutionize humanity's ability to communicate and compute. To pursue these advances, it is mandatory to understand and properly model interactions of light with materials such as silicon and gold at the nanoscale, i.e., the span of a few tens of atoms laid side by side. These interactions are governed by the fundamental Maxwell's equations of classical electrodynamics, supplemented by quantum electrodynamics.This book presents the current state-of-the-art in formulating and implementing computational models of these interactions. Maxwell's equations are solved using the finite-difference time-domain (FDTD) technique, pioneered by the senior editor, whose prior Artech House books in this area are among the top ten most-cited in the history of engineering. You discover the most important advances in all areas of FDTD and PSTD computational modeling of electromagnetic wave interactions.This cutting-edge resource helps you understand the latest developments in computational modeling of nanoscale optical microscopy and microchip lithography. You also explore cutting-edge details in modeling nanoscale plasmonics, including nonlocal dielectric functions, molecular interactions, and multi-level semiconductor gain. Other critical topics include nanoscale biophotonics, especially for detecting early-stage cancers, and quantum vacuum, including the Casimir effect and blackbody radiation.

A time-space spectral method is given for the 1-D nonlinear Maxwell equations. And the spectral method of time multi interval is considered, that is, the interval decomposition is used in time spectral approximation. By computing some numerical examples for 1-D nonlinear Maxwell’s equations, the effectiveness of the proposed method is verified.

We write the charge-free Maxwell equations in a form analogous to that of the Dirac equation for a free electron. This allows us to apply to light some of the ideas developed for the relativistic theory of the electron. Valuable insight is gained, thereby, into the forms of the optical spin and orbital angular momenta.

Discontinuous Galerkin methods are developed for solving the Vlasov–Maxwell system, methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov–Maxwell system. The proposed scheme employs discontinuous Galerkin discretizations for both the Vlasov and the Maxwell equations, resulting in a consistent description of the distribution function and electromagnetic fields. It is proven, up to some boundary effects, that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the Maxwell equations and the underlying approximation spaces. Error estimates are established for several flux choices. The scheme is tested on the streaming Weibel instability: the order of accuracy and conservation properties of the proposed method are verified.