On the theory of reflectivity and transmissivity of a lossless nonlinear dielectric slab

Parameter dependence and stability of guided TE-waves in a lossless nonlinear dielectric slab structure

TE-polarized electromagnetic waves guided by a three-layer structure consisting of a film surrounded by semi-infinite media (all media are characterized by a Kerrlike dielectric function) are investigated by expressing the solution of the field equations in terms of Weierstrass' elliptic functions. Evaluation leads to a universal dispersion relation and its solutions and a universal expression for the power flow. Numerical results are presented for the effective wave number as a function of the intensity of the electric field at the lower surface of the nonlinear film, for various profiles of the field intensity, and for the power flow as a function of the effective wave number.

We develop a method of squared wavefunctions for the vector nonlinear Schrodinger equation. The squared wavefunctions of the octet representation of SU(3) group give periodic solutions in terms of Weierstrass' elliptic functions. Specific limits of the obtained solution are the plane wave, the soliton and cnoidal waves, which were previously obtained using the ansatz of stationary motion.

This paper implements the sub-ODE method and a wide spectrum of solitons are recovered for Kudryashov’s law of refractive index. The self-phase modulation comprises of four nonlinear components of refractive index. The perturbation terms are all of Hamiltonian type and are considered with maximum intensity. The solutions are written in terms of Weierstrass’ elliptic functions and Jacobi’s elliptic function. With the modulus of ellipticity approaching zero or unity, soliton solutions emerge.

Solitons and conservation laws in magneto–optic waveguides with generalized Kudryashov’s equation

Acoustic imaging and sensor modeling are processes that require repeated solution of the acoustic wave equation. Solution of the wave equation can be computationally expensive and memory intensive for large simulation domains. One scheme for speeding up solution of the wave equation is the operator-based upscaling method. The algorithm proceeds in two steps. First, the wave equation is solved for fine grid unknowns internal to coarse blocks assuming the coarse blocks do not need to communicate with neighboring blocks in parallel. Second, these fine grid solutions are used to form a new problem which is solved on the coarse grid. Accurate and efficient wave propagation schemes also must avoid artificial reflections off of the computational domain edges. One popular method for preventing artificial reflections is the nearly perfectly matched layer (NPML) method. In this paper, we discuss applying NPML to operator upscaling for the wave equation. We show that although we only apply NPML to the first step of this two step algorithm (directly affecting the fine grid unknowns only), we still see a significant reduction of reflections back into the domain. We describe three numerical experiments (one homogeneous medium experiment and two heterogeneous media examples) in which we validate that the solution of the wave equation exponentially decays in the NPML regions. Numerical experiments of acoustic wave propagation in two dimensions with a reasonable absorbing layer thickness resulted in a maximum pressure reflection of 3–8%. While the coarse grid acceleration is not explicitly damped in our algorithm, the tight coupling between the two steps of the algorithm results in only 0.1–1% of acceleration reflecting back into the computational domain.

Solitons and conservation laws in magneto-optic waveguides with polynomial law nonlinearity

The equations for the beta relaxation dynamics as obtained within mode-coupling theory for the glass transition are solved asymptotically for parameters near Whitney cusp singularities. The solution is given by a two-parameter scaling law, where the time t enters as ln t and where the scaling times depend exponentially with a Vogel-Fulcher like form on the control parameters. The master function is given in terms of Weierstrass' elliptic function. It describes crossovers from critical relaxation Phi (t) varies as 1/ln2t to a constant f0, to a power-law decay 1/ta, to Phi (t) varies as -lnt, or to Phi (t) varies as ln2t depending on the sector in parameter space. The relaxation data for the Cu-Mn spin-glass alloy can be described by the theory for a time interval of eight decades.

Vacuum structure of the electroweak theory in high magnetic fields

The new approach to the wavelet analysis for the solutions of the homogeneous wave equation in three spatial dimensions is presented. The approach is based on the ideas suggested by G. Kaiser but has different implementation and has some advantages versus the known approach. A new physical wavelet for this wavelet analysis is also presented, with the brief discussion of its main properties. The wavelet analysis has become widely used during the last twenty years and it has a lot of applications nowadays. However most of them are in the field of the numerical processing of the experimental data, digital images, astronomical, geophysical and medical data and other applications of that kind (see, for example, [1], [2]). The amount of the results in the application of the methods of the continuous wavelet analysis to the solutions of the differential equations is not large. In particular the continuous wavelet analysis for the solutions of the three-dimensional homogeneous wave equations with a constant wave speed was first developed by G. Kaiser in his book [3]. He suggests a method for decomposition of the solutions of the wave equation in terms of the localized solutions of the same equation based on the analytic signal transform and on the theory of the analytic functions of several variables. The wavelet for such decomposition was also suggested, and the class of such wavelets was named ’physical wavelets’. The sort of the wavelet analysis developed by Kaiser is close to the holomorphic wavelet transform (see, for example, [2]). However, this approach may be found unfamiliar by the people who deal with the wavelet analysis within the framework of the signal and image processing. The aims of this paper are as follows. First we develop the wavelet analysis for the solutions of the wave equation not involving the analytic signal transform. The ideas, which we base on, were suggested by Kaiser in [3], however their implementation here differs from his approach. The method we use is intrinsic to the common continuous wavelet transform and we hope will be more familiar to the people who work in the area of the signal and image processing. Our approach also provides some advantages in comparison to that, suggested by Kaiser. We enlarge the class of solutions which can be used as the mother wavelets for the analysis. The second aim is to find a new physical wavelet for our method, i.e., to find the solution of the wave equation which will be an admissible wavelet. The new wavelet is constructed by means of the field of point sources and of proxy wavelets using the technique suggested by G. Kaiser. This new spherically symmetric physical wavelet has good properties such as exponential localization in both the coordinate and the Fourier

Complete recurrent algorithms for calculating the higher-order fundamental solutions of covariant linear wave equations for scalar and tensor wave fields on an arbitrary curved spacetime are derived. The higher-order fundamental solutions are the distributions that satisfy the wave equations with the corresponding order covariant derivatives of the Dirac delta function as the source terms. Like the classical Green's function for a scalar wave equation, the higher-order fundamental solutions contain the terms which have support on, and only on, the lightcone as well as tail terms which have support inside the lightcone. With the help of the higher-order fundamental solutions found it is possible to compute the exact multipole solutions of wave equations in a form convenient for practical computations. As applications we consider the exact field of a dipole source of variable strength travelling in an arbitrary curved spacetime and the tail term of scalar multipole waves in the Friedman dust-dominated universe for the case of minimal coupling.

TE-polarized electromagnetic waves, guided by a three-layer slab structure consisting of a central film with quartic permittivity placed between two half spaces with Kerr permittivity, are studied. Traveling-wave solutions of Maxwell's equations are expressed in terms of Weierstrass's elliptic function $\ensuremath{\wp}$. A general dispersion relation is derived and evaluated by using a phase diagram analysis. Emphasis is placed on the conditions of existence and solvability of the dispersion relation. Numerical results are presented.

<abstract><p>Photoacoustic tomography (PAT) is a novel and rapidly developing technique in the medical imaging field that is based on generating acoustic waves inside of an object of interest by stimulating non-ionizing laser pulses. This acoustic wave was measured by using a detector on the outside of the object it was then converted into an image of the human body after several inversions. Thus, one of the mathematical problems in PAT is reconstructing the initial function from the solution of the wave equation on the outside of the object. In this study, we consider the fractional wave equation and assume that the point-like detectors are located on the sphere and hyperplane. We demonstrate a way to recover the initial function from the data, namely, the solution of the fractional wave equation, measured on the sphere and hyperplane.</p></abstract>

On complex soliton solutions, complex elliptic solutions and complex rational function solutions for the Sasa-Satsuma model equation with variable coefficients

Using Lienard-Wiechert fields and the Lorentz Force relation we present self consistent three-dimensional radiationstudies of electron beams moving through periodic electromagnetic structures such as those present in synchrotronsand free-electron laser undulators. Besides providing an economical means of calculating three dimensional vectorradiation fields, our approach yields new insights into individual electron motion as it is driven by both, the velocityfields (Coulomb Fields) and the radiation fields generated by other electrons. We present results ofelectron beam com-pression resulting from longitudinal radiation forces competing in opposition with repulsive velocity field forces. Wediscuss results ofnoiseless three dimensional SelfAmplified Spontaneous Emission (SASE) in the X-Ray region re-sulting from the interaction of a filamentary electron beam with a circularly polarized magnetic undulator. Introduction The usual technique used to study the free-electron laser amplification process entails solving a paraxial version ofthe wave equation at one or more signal frequencies [1,2,3,4]. Although this approach has been quite successful in ex-plaining most features of the free-electron laser (FEL) stimulated emission process, by the nature of its approxima-tions, it has some limitations. For example one disadvantage of working with a paraxial wave equation is that thedescription of the radiation field has incomplete three-dimensional features and consequently only waves having sim-pie spatial structure can be used to solve the FEL problem. Also, working in the frequency domain yields satisfactoryresults only when a few longitudinal modes are needed to deal with the problem. Thus it is impractical to use the fre-quency domain approach when very short electron beam, and consequently short optical pulses, are considered. Per-haps the greatest limitation of present FEL theories is that a solution of the homogeneous wave equation is alwayspresent at the beginning ofthe FEL interaction process. That is, an input signal is introduced so that the so called elec-tron ponderomotive phase is a well defmed quantity. Although this assumption is needed to study the FEL wave am-plification process (stimulated emission) it does not provide sufficient flexibility to study the FEL start-up process.The limitation ofthese theories is even more evident when applied to study coherent radiation effects that occur in self-amplified spontaneous emission (SASE) processes.In this paper we present the derivation of equations of motion for electrons and fields based on Lienard-Wiechertfields, which are exact, three-dimensional solutions of the wave equation in free-space and, as it is done in all othertheories, the electron's motion is governed by the relativistic Newton's Second Law in which the driving force is theLorentz force. The Lorentz force includes fields generated by all electrons in the beam as well as any arbitrary exter-nally prescribed fields. To be more specific we deal with the self-consistent radiation problem of an electron beambeing accelerated by periodic electromagnetic structures (undulators and wigglers), such as those used in synchrotronsand free-electron lasers (FELs). Unlike the restricted solutions of Maxwell's equations obtained by others, theLienard-Wiechert fields are exact, time and space domain free-space solutions of Maxwell's equations for a pointcharge. The attributes ofthese particular solutions allows us to incorporate in a natural way the three dimensional ef-fects of internal forces existing within the electron beam and as a result we can explore easily the three dimensionalnature ofthe radiated fields. This approach becomes particularly useful in dealing with the FEL start-up problem aswell as when studying non-stimulated coherent radiation effects. Unlike the approach used by other, our scheme re-quires no initial artificial electromagnetic seed to start the numerical solution of the problem. Furthermore becauseLienard-Wiechert fields are time-domain solutions of the wave equation, we can study non-periodic electron beam sys-tems. In particular we can deal satisfactorily with three-dimensional effects of very long and very short electron bunch-es. As an example of the power of our approach we present results of three dimensional radiation features of self-490/SPIE Vol. 2522