1. Maxwell’s equations 1 1.1. General case 2 1.2. Maxwell equations in integral form 3 1.3. Boundary conditions at a surface of discontinuity 4 1.4. Maxwell’s equations in the absence of charges 7 1.5. The wave equation 7 1.6. Dispersion equation 8 1.7. Plane waves 8 2. Fresnel formulas 9 2.1. Rewriting the Fresnel Equations 12 2.2. The Poynting vector 12 2.3. Polarization 13 2.4. Estimation of...

We explore the existence of solitons in a nonlinear, dispersive, amplifying medium based on a model that makes neither the parabolic-gain approximation nor the rate-equation approximation. Without these approximations, the Maxwell-Bloch equations no longer reduce to a Ginzburg-Landau equation and do not appear to have analytic soliton solutions. We use numerical simulations to show that solitar...

In this work we investigate collective excitations at the boundary of a recently constructed 4D quantum Hall state. Local bosonic operators for creating these collective excitations can be constructed explicitly. Massless relativistic wave equations with helicity S can be derived exactly for these operators from their Heisenberg equation of motion. For the S = 1 and S = 2 cases these equations ...

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole-Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov-Novikov equation for the refractive index. It is demonstrated that the Veselov-Novikov hierarchy is ...

In this paper, we derive regularity estimates for the electric field integral operator which arises when formulating the time-harmonic Maxwell problem as a boundary integral equation. More precisely, we show that the regularity constants can be bounded polynomially in terms of the frequency, where the degree of the polynomial depends on the regularity order and is given explicitly. The paper co...

Hydrodynamic equations for an inelastic Maxwell model are derived from the inelastic Boltzmann equation based on a systematic Chapman-Enskog perturbative scheme. Transport coefficients appear in Navier-Stokes order have been determined as a function of the restitution coefficient e, which cannot be defined for small e as a result of the high energy tail of the velocity distribution function obe...

The essence of the boundary-field equation method is the reduction of the boundary value problem under consideration to an equivalent nonlocal boundary value problem in a bounded domain by using boundary integral equations. The latter can then be treated by the standard variational method including its numerical approximations. In this paper, various formulations of the nonlocal boundary value ...

We discuss the grazing collision limit of certain kinetic models of Bose-Einstein particles obtained from a suitable modification of the onedimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a nonlinear Fokker-Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in [13]. In order to do so, we perform a study of t...

We study discretizations of the Maxwell-Klein-Gordon equation as an example of a constrained geometric non-linear evolution partial differential equation. For the temporal gauge we propose a fully discrete scheme which preserves the non-linear constraint thanks to a special application of Lagrange multipliers. We show that the method generalizes to Hamiltonian wave equations whose kinetic and p...