Nonlinear Maxwell Equations Research Articles

A fourth order Runge-Kutta type of exponential time differencing and triangular spectral element method for two dimensional nonlinear Maxwell's equations

In this paper, we study a numerical scheme to solve the nonlinear Maxwell's equations. The discrete scheme is based on the triangular spectral element method (TSEM) in space and the exponential time differencing fourth-order Runge-Kutta (ETDRK4) method in time. TSEM has the advantages of spectral accuracy and geometric flexibility. The ETD method involves exact integration of the linear part of the governing equation followed by an approximation of an integral involving the nonlinear terms. The RK4 scheme is introduced for the time integration of the nonlinear terms. The stability region of the ETDRK4 method is depicted. Moreover, the contour integral in the complex plan is utilized and improved to compute the matrix function required by the implementation of ETDRK4. The numerical results demonstrate that our proposed method is of exponential convergence with the order of basis function in space and fourth order accuracy in time.

Multiple cylindrically symmetric solutions of nonlinear Maxwell equations

In this paper, we study the following nonlinear time-harmonic Maxwell equations \begin{equation}\label{equation 0.1} \nabla\times(\nabla \times E)-\omega^2\varepsilon(x)E =P(x)|E|^{p-2}E+Q(x)|E|^{q-2}E, \end{equation} where $\varepsilon(x)$ is the permittivity of the material, $x\in\mathbb{R}^{3}$, $1< q< {p}/({p-1})< 2< p< 6$, $P(x),Q(x)\in C\left(\mathbb{R}^{3},\mathbb{R}\right)$. Under some special cylindrical symmetric conditions for $\varepsilon(x)$, $P(x)$ and $Q(x)$, we obtain infinite many cylindrically symmetric solutions of \eqref{equation 0.1} by using variational method and fountain theorems without $\tau$-upper semi-continuity.

Well-posedness and exponential stability of nonlinear Maxwell equations for dispersive materials with interface

In this paper we consider an abstract Cauchy problem for a Maxwell system modeling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain.

Influence of rippled density and laser profile on third harmonic generation using cosh‐Gaussian laser pulses in inhomogeneous magnetized plasmas

AbstractThe third harmonic generation of cosh‐Gaussian laser pulses in inhomogeneous magnetized plasmas is investigated. The plasma is considered inhomogeneous with a sinusoidal form, and the external magnetic field has an oscillating wiggler structure. Various ponderomotive forces are obtained considering the external magnetic field, the magnetic field of the laser beam, and different orders of electron velocity. These forces are nonlinear, which leads to a nonlinear current density. Using linear and nonlinear current densities and Maxwell's equations, the wave equation of the third harmonic wave is obtained. Results show that by increasing the deviation parameter from the center and the external magnetic field, the third harmonic electric field increases. It is also indicated that by increasing the laser beam width, the excited electric field decreases. Finally, it is found that by increasing the wavenumber and amplitude of the rippled electron density, the amplitude of the third harmonic electric field increases.

Positive singular solutions of a nonlinear Maxwell equation arising in mesoscopic electromagnetism

At least one positive radial singular solution and infinitely many positive non-radial singular solutions of a nonlinear Maxwell equation in dimension two are constructed in a refined local version. The nonlinear Maxwell equation arises in mesoscopic electromagnetism. To construct the singular solutions, we need to know more detailed asymptotic behavior at the isolated singular point of the prescribed singular solutions than those already known. We will establish the asymptotic expansion up to arbitrary orders at the isolated singular point of a prescribed singular solution of the nonlinear Maxwell equation. Using such expansions, we construct positive singular solutions via fixed point arguments in some weighted Hölder spaces.

An energy-stable finite element method for nonlinear Maxwell's equations

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell's equations in the optical media characterized by linear Lorentz dispersion, nonlinear Kerr effect and delayed Raman scattering. Using Crank-Nicolson time discretization and special treatment of the nonlinearity, we propose a second order accurate (in time) finite element method for the nonlinear problem. Fully discrete energy stability analysis is established to show that the scheme is unconditionally stable. The nonlinear system is solved by Newton-Krylov method. In order to efficiently solve the linearized algebraic system in each Newton sub-iteration, we further develop an effective preconditioner by using suitable approximations to the nonlinearity and the auxiliary space preconditioning technique. Numerical experiments are provided to examine the accuracy, the energy stability, the parallel scalability and the robustness in preconditioning of our proposed scheme. We then apply the scheme to simulate the spatial soliton propagation in 3D nonlinear glass. We also demonstrate the performance of the scheme in handling complex geometry through simulating the wave scattering by a spherical hole in the nonlinear optical glass.

Time-space Spectral Method for the 1-D nonlinear Maxwell Equations

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.

Frequency-Based Estimation of the B-H Curve of Steel Strips in a Continuous Induction Furnace*

A parameter estimation strategy for continuous inductive heating of ferromagnetic steel strips is developed and investigated. During strip processing in an induction furnace, the parameters of the magnetization curve (B-H curve) of the strip material are identified by a moving horizon estimator (MHE). This estimator uses measurements of the resonance frequency of the electric circuit. A tailored 2D harmonic model of the electromagnetic subsystem is used to calculate the complex coil impedance. Other model parameters are identified in a scenario, where the magnetization curve of the strip is known. The approach does not require a thermal model of the induction furnace in favor of the computing time and the achieved accuracy. For model validation, the simulated resonance frequency is compared with measurements. To avoid a high computational effort, the effective magnetization approach is applied to approximate the solution of the nonlinear Maxwell equations. The effective B-H curve is mathematically expressed in terms of a linear and an exponential function and parametrized by two parameters. The MHE is validated in a test scenario using a strip with a known magnetization curve.

Oscillations of highly magnetized non-rotating neutron stars

Highly magnetized neutron stars are promising candidates to explain some of the most peculiar astronomical phenomena, for instance, fast radio bursts, gamma-ray bursts, and superluminous supernovae. Pulsations of these highly magnetized neutron stars are also speculated to produce detectable gravitational waves. In addition, pulsations are important probes of the structure and equation of state of the neutron stars. The major challenge in studying the pulsations of highly magnetized neutron stars is the demanding numerical cost of consistently solving the nonlinear Einstein and Maxwell equations under minimum assumptions. With the recent breakthroughs in numerical solvers, we investigate pulsation modes of non-rotating neutron stars which harbour strong purely toroidal magnetic fields of 1015−17 G through two-dimensional axisymmetric general-relativistic magnetohydrodynamics simulations. We show that stellar oscillations are insensitive to magnetization effects until the magnetic to binding energy ratio goes beyond 10%, where the pulsation mode frequencies are strongly suppressed. We further show that this is the direct consequence of the decrease in stellar compactness when the extreme magnetic fields introduce strong deformations of the neutron stars.

Trade‐Off between Second‐ and Third‐Order Nonlinearities, Ultrafast Free Carrier Absorption and Material Damage in Silicon Nanoparticles

AbstractReaching the optimal second‐ and third‐order nonlinear conversion efficiencies while avoiding undesirable free carrier absorption losses and material damage in ultrashort laser‐excited nanostructures is a challenging obstacle in all‐dielectric ultrafast nanophotonics. In order to elucidate the main aspects of this problem, a multi‐physical model is developed, coupling nonlinear Maxwell equations supplied by surface and bulk nonlinearities with free carrier hydrodynamic equations for electron–hole plasma kinetics and electron‐ion transfer for silicon. The maximum feasible efficiencies for a single spherical particle supporting different electric and magnetic resonances are compared, and the harmonic yields are further optimized by tuning lattice resonances in a periodic arrangement of nanoparticles. Results support the dominant role of magnetic dipole and quadrupole contributions in the enhancement of the third harmonic and the electric dipole for the second harmonic, as well as the possibility to further improve the conversion of both harmonics simultaneously at least by two orders of magnitude by designing properly the resonant metasurface.

Infinitely many cylindrically symmetric solutions of nonlinear Maxwell equations with concave and convex nonlinearities

Infinitely many cylindrically symmetric solutions of nonlinear Maxwell equations with concave and convex nonlinearities

Traveling waves for a quasilinear wave equation

We consider a 2+1 dimensional wave equation appearing in the context of polarized waves for the nonlinear Maxwell equations. The equation is quasilinear in the time derivatives and involves two material functions V and Γ. We prove the existence of traveling waves which are periodic in the direction of propagation and localized in the direction orthogonal to the propagation direction. Depending on the nature of the nonlinearity coefficient Γ we distinguish between two cases: (a) Γ∈L∞ being regular and (b) Γ=γδ0 being a multiple of the delta potential at zero. For both cases we use bifurcation theory to prove the existence of nontrivial small-amplitude solutions. One can regard our results as a persistence result which shows that guided modes known for linear wave-guide geometries survive in the presence of a nonlinear constitutive law. Our main theorems are derived under a set of conditions on the linear wave operator. They are subsidized by explicit examples for the coefficients V in front of the (linear) second time derivative for which our results hold.

Nonlinear Maxwell equations and strong-field electrodynamics

We investigate two Lagrangian models of nonlinear electrodynamics (NLED). These models lead to two different sets of nonlinear (NL) Maxwell equations. The first case deals with the well-known Heisenberg-Euler (HE) model of electromagnetic (EM) self-interactions in a vacuum where only the lowest orders in EM Lorentz invariants are considered. The second instance proposes an extension of the HE model. It consists of a NL Maxwell-Dirac spinor model where the EM field modifies the dynamics of the energy-momentum operator sector of the Dirac Lagrangian instead of its rest-mass term counterpart. This work complements our recent research on NL Dirac equations in the strong EM field regime.

An Energy-Stable Finite Element Method for Nonlinear Maxwell's Equations

An Energy-Stable Finite Element Method for Nonlinear Maxwell's Equations

Long-time behavior of a nonlinear electromagnetic wave in vacuum beyond the linear approximation

A quantum nature of vacuum is expected to affect electromagnetic fields in vacuum as a nonlinear correction, yielding nonlinear Maxwell's equations. We extend the finite-difference time-domain (FDTD) method in the case that the nonlinear electromagnetic Lagrangian is quartic with respect to the electric field and magnetic flux density. With this extension, the nonlinear Maxwell's equations can be numerically solved without making any assumptions on the electromagnetic field. We demonstrate examples of self-modulations of nonlinear electromagnetic waves in a one-dimensional cavity, in particular, in a timescale beyond an applicable range of linear approximation. A momentarily small nonlinear correction can accumulate and a comparably large self-modulation can be achieved in a long timescale even though the electromagnetic field is not extremely strong. Further, we analytically approximate the nonlinear electromagnetic waves in the cavity and clarify the characteristics, for example, how an external magnetic flux density changes the self-modulations of phase and polarization.

Inverse problems for nonlinear Maxwell's equations with second harmonic generation

In the current paper we consider an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process. We show the unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium by making electromagnetic measurements on the boundary. We are interested in the case when a frequency is fixed.

A time-domain finite element scheme and its analysis for nonlinear Maxwell's equations in Kerr media

The purpose of this paper is to develop and analyze a finite element method for solving the time-dependent Maxwell's equations in nonlinear Kerr media. The proposed fully-discrete scheme is proved to be conditionally stable and optimally convergent in the spatial variable and second order in time. Numerical results are presented to support our theoretical analysis and also to demonstrate the practical soliton propagation phenomena in Kerr media.

A space-time spectral method for the 1-D Maxwell equation

<abstract> A Legendre-tau space-time spectral method is established for the 1-D Maxwell equation. The polynomials of different degrees are used to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. Also, the time multi-interval Legendre-tau space-time spectral method is considered to keep the long-time computation stable. Error estimates for the method of single and multi-internal are given, respectively. Moreover, the space-time spectral method is applied to the numerical solutions of the 1-D nonlinear Maxwell equation and describes its implicit-explicit iteration scheme. Numerical examples are compared with some other methods, which verifies the effectiveness of the methods for the 1-D Maxwell equation. </abstract>

Solutions to a nonlinear Maxwell equation with two competing nonlinearities in $\mathbb{R}^3$

We are interested in the nonlinear, time-harmonic Maxwell equation $$ \nabla \times (\nabla \times \mathbf{E} ) + V(x) \mathbf{E} = h(x, \mathbf{E})\mbox{ in } \mathbb{R}^3 $$ with sign-changing nonlinear term $h$, i.e. we assume that $h$ is of the form $$ h(x, \alpha w) = f(x, \alpha) w - g(x, \alpha) w $$ for $w \in \mathbb{R}^3$, $|w|=1$ and $\alpha \in \mathbb{R}$. In particular, we can consider the nonlinearity consisting of two competing powers $h(x, \mathbf{E}) = |\mathbf{E}|^{p-2}\mathbf{E} - |\mathbf{E}|^{q-2}\mathbf{E}$ with $2 < q < p < 6$. Under appriopriate assumptions, we show that weak, cylindrically equivariant solutions of the special form are in one-to-one correspondence with weak solutions to a Schr\"odinger equation with a singular potential. Using this equivalence result we show the existence of the least energy solution among cylindrically equivariant solutions of the particular form to the Maxwell equation, as well as to the Schr\"odinger equation.

Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type media

In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.