Maxwell System Research Articles

Charged Shear-Free Fluids and Complexity in First Integrals.

The equation is the charged generalization of the Emden-Fowler equation that is crucial in the study of spherically symmetric shear-free spacetimes. This version arises from the Einstein–Maxwell system for a charged shear-free matter distribution. We integrate this equation and find a new first integral. For this solution to exist, two integral equations arise as integrability conditions. The integrability conditions can be transformed to nonlinear differential equations, which give explicit forms for and in terms of elementary and special functions. The explicit forms and arise as repeated roots of a fourth order polynomial. This is a new solution to the Einstein-Maxwell equations. Our result complements earlier work in neutral and charged matter showing that the complexity of a charged self-gravitating fluid is connected to the existence of a first integral.

Can a single scalar second order PDE govern well the propagation of the electric wave field in a heterogeneous 3D medium?

Abstract It is demonstrated in this paper that the propagation of the electric wave field in a heterogeneous medium in 3D can sometimes be governed well by a single scalar second order PDE, which is derived from Maxwell’s equations. The corresponding component of the electric field dominates two other components. This justifies some past results of the second author with coauthors about numerical solutions of coefficient inverse problems with experimental electromagnetic data. In addition, since it is simpler to work in applications with a single scalar second order PDE rather than with the complete Maxwell system, the result of this paper might be useful to researchers working on applied issues of the propagation of electromagnetic waves in inhomogeneous media.

Cloaking Property of a Plasmonic Structure in Doubly Complementary Media and Three-Sphere Inequalities with Partial Data

We investigate cloaking property of negative-index metamaterials in the time-harmonic electromagnetic setting for the so-called doubly complementary media. These are media consisting of negative-index metamaterials in a shell (plasmonic structure) and positive-index materials in its complement for which the shell is complementary to a part of the core and a part of the exterior of the core-shell structure. We show that an arbitrary object is invisible when it is placed close to a plasmonic structure of a doubly complementary medium as long as its cross section is smaller than a threshold given by the property of the plasmonic structure. To handle the loss of the compactness and of the ellipticity of the modeling Maxwell equations with sign-changing coefficients, we first obtain Cauchy's problems associated with two Maxwell systems using reflections. We then derive information from them, and combine it with the removing localized singularity technique to deal with the localized resonance. A central part of the analysis on the Cauchy's problems is to establish three-sphere inequalities with partial data for general elliptic systems, which are interesting in themselves. The proof of these inequalities first relies on an appropriate change of variables, inspired by conformal maps, and is then based on Carleman's estimates for a class of degenerate elliptic systems.

Existence and concentration of positive solutions for Klein–Gordon–Maxwell system with asymptotically linear nonlinearities

In this paper, we are concerned with the Klein–Gordon–Maxwell system −Δu + (λa(x) + 1)u − (2ω + ϕ)ϕu = f(x, u) in R3 and Δϕ = (ω + ϕ)u2 in R3, where u and ϕ are unknowns and f is an asymptotically linear term. We prove the existence of positive solutions based on variational methods and some analytical techniques. Furthermore, the properties of decay estimates and asymptotic behavior for the positive solution are studied here.

Higher-Order Accurate and Conservative Hybrid Numerical Scheme for Relativistic Time-Fractional Vlasov-Maxwell System

The historical analysis demonstrates that plasma scientists produced a variety of numerical methods for solving “kinetic” models, i.e., the Vlasov-Maxwell (VM) system. Still, on the other hand, a significant fact or drawback of most algorithms is that they do not preserve conservation philosophies. This is a crucial fact that cannot be disregarded since the Vlasov Maxwell system is associated with conservation rules and is capable of assessing after the accomplishment of certain helpful mathematical actions. To examine the fractional-order routine of charged particles, we constructed a fractional-order plasma model and proposed a higher-order conservative numerical approach based on operational matrices theory and Shifted Gegenbauer estimations. Numerical convergence is investigated to confirm its competence and compatibility. This concept may be used in problems involving variable order and multidimensionality, such as those involving Vlasov and Boltzmann systems.

(2 + 1)-Maxwell Equations in Split Quaternions

The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two SO(2,1) subgroups of the full group of symmetry of the norms of split quaternions, SO(2,2). One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second SO(2,1) subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.

Homogenization of Maxwell’s equations and related scalar problems with sign-changing coefficients

In this work, we are interested in the homogenization of time-harmonic Maxwell's equations in a composite medium with periodically distributed small inclusions of a negative material. Here a negative material is a material modelled by negative permittivity and permeability. Due to the sign-changing coefficients in the equations, it is not straightforward to obtain uniform energy estimates to apply the usual homogenization techniques. The goal of this article is to explain how to proceed in this context. The analysis of Maxwell's equations is based on a precise study of two associated scalar problems: one involving the sign-changing permittivity with Dirichlet boundary conditions, another involving the sign-changing permeability with Neumann boundary conditions. For both problems, we obtain a criterion on the physical parameters ensuring uniform invertibility of the corresponding operators as the size of the inclusions tends to zero. In the process, we explain the link existing with the so-called Neumann-Poincare operator, complementing the existing literature on this topic. Then we use the results obtained for the scalar problems to derive uniform energy estimates for Maxwell's system. At this stage, an additional difficulty comes from the fact that Maxwell's equations are also sign-indefinite due to the term involving the frequency. To cope with it, we establish some sort of uniform compactness result.

Black Hole and Wormhole Solutions in Einstein–Maxwell Scalar Theory

We classified and studied the charged black hole and wormhole solutions in the Einstein–Maxwell system in the presence of a massless, real scalar field. The possible existence of charged black holes in general scalar–tensor theories was studied in Bronnikov et al., 1999; black holes and wormholes exist for a negative kinetic term for the scalar field. Using a conformal transformation, the static, spherically symmetric possible structures in the minimal coupled system are described. Besides wormholes and naked singularities, only a restricted class of black hole exists, exhibiting a horizon with an infinite surface and a timelike central singularity. The black holes and wormholes defined in the Einstein frame have some specificities with respect to the non-minimal coupling original frame, which are discussed in the text.

Atangana-Baleanu Caputo fractional-order modeling of plasma particles with circular polarization of LASER light: An extended version of Vlasov-Maxwell system

In this study, we design a specific geometry of plasma particles and further translate it into mathematical form, i.e., formulated time-fractional semi relativistic Vlasov Maxwell system. This extended version of model is very significant and ground-breaking because it has ability to study the behavior of plasma particles at macroscopic and microscopic time-evaluation scales, which is not studied yet. This model is further tackled with numerical strategy, projected in accordance with spectral and finite difference approximations. The numerical results demonstrate that there are certain variations of the plasma particles that were out of sight at fractional scale and dynamically, we have entered into the real profundity of the problem. Numerical convergence of the projected technique is also inspected. These designed tools, such as the plasma and numerical methods, are very significant and can say that it is a remarkable contribution in this field. In addition, the established numerical structure can utilize to scrutinize the required numerical solution of highly non-linear and fractional problems.

Operator-norm homogenisation estimates for the system of Maxwell equations on periodic singular structures

For arbitrarily small values of varepsilon >0, we formulate and analyse the Maxwell system of equations of electromagnetism on varepsilon -periodic sets S^varepsilon subset {{mathbb {R}}}^3. Assuming that a family of Borel measures mu ^varepsilon , such that mathrm{supp}(mu ^varepsilon )=S^varepsilon , is obtained by varepsilon -contraction of a fixed 1-periodic measure mu , and for right-hand sides f^varepsilon in L^2({mathbb {R}}^3, dmu ^varepsilon ), we prove order-sharp norm-resolvent convergence estimates for the solutions of the system. Our analysis includes the case of periodic “singular structures”, when mu is supported by lower-dimensional manifolds. The estimates are obtained by combining several new tools we develop for analysing the Floquet decomposition of an elliptic differential operator on functions from Sobolev spaces with respect to a periodic Borel measure. These tools include a generalisation of the classical Helmholtz decomposition for L^2 functions, an associated Poincaré-type inequality, uniform with respect to the parameter of the Floquet decomposition, and an appropriate asymptotic expansion inspired by the classical power series. Our technique does not involve any spectral analysis and does not rely on the existing approaches, such as Bloch wave homogenisation or the spectral germ method.

A semi‐relativistic time‐fractional Vlasov–Maxwell code for numerical simulation based on circular polarization and Landau damping instability

AbstractThe Vlasov–Maxwell system is a revolutionary tool to model and analyze the dynamic behavior of the collisionless plasma in the existence of the electromagnetic field. In this perspective, the investigation of this system with the concept of the time‐fractional derivative is the new benchmark and also one of the main objectives of this article. In order to construct the numerical solution, we propose and implement a novel scheme which grounded on finite‐difference and spectral approximations. The temporal variable is tackled by using finite‐difference approximation, whereas spatial variables approximated via sifted Gegenbauer polynomials. Various simulations are performed to explore new phenomena and demonstrate the accuracy and reliability of the proposed method. Convergence and error bound of the suggested methods are investigated theoretically, whereas stability analysis performed numerically. Moreover, the developed method can be used conveniently to examine the numerical solution of other multidimensional highly nonlinear fractional or variable order problems of physical nature.

Multi-Scale Paraxial Models to Approximate Vlasov–Maxwell Equations

Abstract Even today, solving numerically the time-dependent Vlasov–Maxwell equations is a challenging issue, and developing simpler but accurate approximate models is still worthwhile. Here, we propose a new family of paraxial asymptotic models that approximates the Vlasov–Maxwell system of equations. We introduce parameters in our models that allow us to handle relativistic cases, much slower beams or even non-relativistic cases. These models are derived by introducing a small parameter and provide static or quasi-static approximate equations that are 𝑛-th order accurate; 𝑛 may be chosen as required. Practically, one can select a model by determining the regime one is interested in and choosing the degree of accuracy needed.

A linearised inverse conductivity problem for the Maxwell system at a high frequency

We consider a linearised inverse conductivity problem for electromagnetic waves in a three dimensional bounded domain at a high time-harmonic frequency. Increasing stability bounds for the conductivity coefficient in the full Maxwell system and in a simplified transverse electric mode are derived. These bounds contain a Lipschitz term with a factor growing polynomially in terms of the frequency, a Hölder term, and a logarithmic term which decays with respect to the frequency as a power. To validate this increasing stability numerically, we propose a reconstruction algorithm aiming at the recovery of sufficiently many Fourier modes of the conductivity. A numerical evidence shed light on the influence of the growing frequency and confirms the improved resolution.

Structure-preserving discretization of Maxwell's equations as a port-Hamiltonian system

This work demonstrates the discretization of the boundary-controlled Maxwell equations, recast as a port-Hamiltonian system (pHs). After a reminder on the Stokes-Dirac structure associated with the Maxwell system, we introduce different partitioned weak formulations that preserve the pHs structure, and its associated power balance, at the semi-discrete level. These weak formulations are compared through numerical applications to closed non-perfectly conducting cavities and open waveguides under transverse approximation.

Propagation of Regularity and Long Time Behavior of the 3D Massive Relativistic Transport Equation II: Vlasov–Maxwell System

Given any smooth, suitably small initial data, which decays polynomially at infinity, we prove global regularity for the 3D relativistic massive Vlasov–Maxwell system. In particular, the compact support assumption, which was widely used in the literature, is not imposed on the initial data. Our proofs are based on a combination of the Klainerman vector field method and the Fourier method, which allows us to exploit a crucial hidden null structure in the relativistic Vlasov–Maxwell system.

Weak Wavefront Solutions of Maxwell’s Equations in Conducting Media

We analyze the propagation of electromagnetic fronts in unbounded electric conductors. Our analysis is based on the Maxwell model of electromagnetism that includes the displacement current and Ohm’s law in its simplest forms. A weak electromagnetic front is a propagating interface at which the electromagnetic field remains continuous while its first- and higher-order derivatives experience finite jump discontinuities. Remarkably, analysis of such fronts can be performed autonomously, i.e. strictly in terms of the quantities defined on the front. This property opens the possibility of establishing exact analytical solutions of the exact Maxwell system along with the evolution of the front.

Asymptotic behavior of eigenvalues of the Maxwell system in the presence of small changes in the interface of an inclusion

<p style='text-indent:20px;'>In this paper, we derive rigorously asymptotic formulas for perturbations in the eigenfrequencies of a Maxwell system due to small changes in the interface of a smooth inclusion. Taking advantage of small perturbations, we use a rigorous asymptotic analysis to develop an asymptotic formula for the case where the eigenvalue of the reference problem is simple or multiple. We show that our asymptotic formulas can be expressed in terms of the electric permittivity and the profile function <inline-formula><tex-math id="M1">\begin{document}$ h $\end{document}</tex-math></inline-formula> modelling the shape perturbation. We assume that our results are ambitious tools to solve the inverse problem of identifying interface changes (deformations) of inclusions, given eigenvalues measurements.</p>

Existence of positive ground state solutions of critical nonlinear Klein–Gordon–Maxwell systems

In this paper we study the following nonlinear Klein–Gordon–Maxwell system { − Δ u + [ m 0 2 − ( ω + φ ) 2 ] u = f ( u ) in R 3 , Δ φ = ( ω + φ ) u in R 3 , where 0 < ω < m 0 . Based on an abstract critical point theorem established by Jeanjean, the existence of positive ground state solutions is proved, when the nonlinear term f ( u ) exhibits linear near zero and a general critical growth near infinity. Compared with other recent literature, some different arguments have been introduced and some results are extended.

Fractional Moisil-Teodorescu operator in elasticity and electromagnetism

In this work, we introduce a fractional generalization of the classical Moisil-Teodorescu operator that provides a concise notation for presenting a mathematical formulation of physical systems in fractional space from various branches of science and engineering. The method used in this article, called the Stillinger’s formalism, is combined in a novel way with quaternionic analysis. As a concrete application, a quaternionic reformulation of a fractional time-harmonic Maxwell system is established, thus showing a deep relation between it solutions with those of the perturbed fractional Moisil-Teodorescu operator. Furthermore, the fractional constructions here will find further applications in several applied research areas such as in hydrodynamics and magneto-hydrodynamics.

Fractional order plasma modeling based on linear polarization of LASER light: an Atangana–Baleanu Caputo approach

PurposeThe purpose of this offered research is to articulate a multifaceted kind of highly unstable initial perturbation and further analyze the performance of the plasma particles for time-fractional order evaluation.Design/methodology/approachFor this purpose, the authors designed specific geometry and further interpreted it into the mathematical model using the concepts of the Vlasov Maxwell system. The suggested algorithm is based on the finite-difference and spectral estimation philosophy. The management of time and memory in generic code for computational purposes is also discussed.FindingsThe main purpose is to analyze the fractional behavior of plasma particles and also the capability of the suggested numerical algorithm. Due to initial perturbations, there are a lot of sudden variations that occurred in the formulated system. Graphical behavior shows that SR parameter produces devastation as compared to others. The variation of fractional parameter between the defend domain demonstrates the hidden pictures of plasma particles. The design scheme is efficient, convergent and has the capability to cover the better physics of the problem.Practical implicationsPlasma material is commonly used in different areas of science. Therefore, in this paper, the authors increase the capability of the mathematical plasma model with specific geometry, and further suitable numerical algorithm is suggested with detailed physical analysis of the outcomes. The authors gave a new direction to study the performance of plasma particles under the influence of LASER light.Originality/valueIn the recent era, science has produced a lot of advancements to study and analyze the physical natural process, which exist everywhere in the real word. On behalf of this current developments, it is now insufficient to study the first-order time evaluation of the plasma particles. One needs to be more precise and should move toward the bottomless state of it, that is, macroscopic and microscopic time-evaluation scales, and it is not wrong to say that there exits a huge gap, to study the time evaluation in this discussed manner. The presented study is entirely an advanced and efficient way to investigate the problem into the new directions. The capability of the proposed algorithm and model with fractional concepts can fascinate the reader to extend to the other dimensions.