On the inverse scattering approach and action-angle variables for the Dullin-Gottwald-Holm equation

The Poisson brackets for the scattering data of the Dullin-Gottwald-Holm equation are computed. Then, the action-angle variables are expressed in terms of the scattering data. We also discuss on the inverse scattering problem for this equation.     

An inverse model for a free-boundary problem with a contact line: Steady case

This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase. It is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated. We describe an efficient iterative solution method for the Stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional. Performance of the proposed approach is illustrated with sample computations concerning 2D steady solidification phenomena.     

Determination of the electromagnetic properties in magnetic composite materials by inverse homogenisation

The paper proposes a numerical procedure for evaluating the geometrical and electromagnetic properties of the microstructure of a heterogeneous medium starting from the material effective characteristics. The identification process is developed by applying an inverse homogenisation technique, based on multiple scale expansion theory. The Newton–Raphson scheme and the Rosenbrock algorithm are tested. The efficiency of the second approach is proved by reporting several numerical results.     

Fractional vector calculus and fractional Maxwell’s equations

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral vector operations. The fractional Green’s, Stokes’ and Gauss’s theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential calculus of differential forms is discussed. Fractional nonlocal Maxwell’s equations and the corresponding fractional wave equations are considered.     

Numerical solution for an inverse scattering problem of non-periodic rough surfaces

The paper presents the first results of a numerical study of inverse diffraction devoted to non-periodic rough surfaces in optics. Two kinds of rough surfaces are considered: first gratings with a finite number of grooves, and second random rough surfaces. For shallow surfaces, adequate Fourier theories have been employed with success. On the other hand, for deeper asperities, rigorous methods are needed and generally, the reconstruction of the profile is more difficult. For both Fourier and rigorous methods, the limit of resolution is studied numerically and numerous examples of reconstruction are given.Instituto Politecnico National, Escuela Superior de Fisica y Matematicas, Mexico, D.F., Mexico     

Total reflection of electromagnetic waves propagating from an isotropic medium to an indefinite metamaterial

The existence conditions for total reflection and the corresponding critical angle at the interface separating an isotropic medium and an indefinite metamaterial for TE- and TM-polarized electromagnetic waves are obtained. For different kinds of indefinite metamaterial, there appear different total reflection phenomena. Particularly, the anomalous total reflection in which the incident angle is smaller than the critical angle and the Brewster’s angle can be smaller than the critical angle can occur for anti-cutoff medium. Furthermore, the omnidirectional total reflection exists for the always cutoff medium and anti-cutoff medium. The total reflection depends on the thickness of indefinite metamaterial when the indefinite metamaterial is finite, and the photon tunneling phenomenon can occur when the thickness of indefinite metamaterial is smaller than wavelength.     

Inverse statistical problems: from the inverse Ising problem to data science

Inverse problems in statistical physics are motivated by the challenges of ‘big data’ in different fields, in particular high-throughput experiments in biology. In inverse problems, the usual procedure of statistical physics needs to be reversed: Instead of calculating observables on the basis of model parameters, we seek to infer parameters of a model based on observations. In this review, we focus on the inverse Ising problem and closely related problems, namely how to infer the coupling strengths between spins given observed spin correlations, magnetizations, or other data. We review applications of the inverse Ising problem, including the reconstruction of neural connections, protein structure determination, and the inference of gene regulatory networks. For the inverse Ising problem in equilibrium, a number of controlled and uncontrolled approximate solutions have been developed in the statistical mechanics community. A particularly strong method, pseudolikelihood, stems from statistics. We also review the inverse Ising problem in the non-equilibrium case, where the model parameters must be reconstructed based on non-equilibrium statistics.     

Exact solutions of semiclassical non-characteristic Cauchy problems for the sine-Gordon equation

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter ε tends to zero. Assuming natural initial data having the profile of a moving −2π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all ε>0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small ε. This sequence of exact solutions is analogous to that of the well-known N-soliton (or higher-order soliton) solutions of the focusing nonlinear Schrödinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit ε↓0 one observes the appearance of nonlinear caustics, i.e. curves in space-time that are independent of ε but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases.In the appendices, we give a self-contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic L¹-Sobolev initial data, and Appendix B establishes the well-posedness for L^p-Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).     

Studies on magnetic photonic band-gap material at microwave frequency

The microwave transmission characteristics of magnetic photonic band-gap (MPBG) materials, in which the periodic structures are composed of alternating layers of polycarbonate with and without ferromagnetic nanowires, are studied. We present a theoretical method to investigate the transmission spectra of the MPBG material. The band-gap effect varies with the periodic parameters of the MPBG structure. Our calculation can describe well the experimental measurement value. The influence of the applied static magnetic field on the MPBG and the ferromagnetic resonance phenomenon has been studied.     

Effects of metal and “magnetic wall” on the dispersion characteristic of magnetostatic waves

The dispersion relation of magnetostatic waves tangentially magnetized to saturation ferrite film, with a “magnetic wall” condition (tangential component of microwave magnetic field is equal to zero) on one of the film surface and with a metal condition on the opposite surface is analyzed. The dispersion characteristics show that unidirectional magnetostatic waves appear in this structure: they can transfer energy in one direction only and fundamentally cannot transfer energy in the opposite direction. The dispersion-free propagation of magnetostatic waves also is possible in the structure in a wide frequency interval.     

Light scattering and transmission spectra of the Menger sponge fractal

Selection rules for light scattering and transmission by the Menger sponge fractal were derived by the group theory based on the symmetry of its localized electromagnetic eigenmodes. The light scattering and transmission spectra were calculated by the finite-difference time-domain method and compared with the eigenfrequencies of the localized modes obtained from the dipole radiation spectra. Their correspondence is quite good and supports the accuracy of the numerical calculation and the correctness of the selection rules.     

Numerical Simulation of Electromagnetic Waves Scattering by Discrete Exterior Calculus

We show how to construct discrete Maxwell equations by discrete exterior calculus. The new scheme has many virtues compared to the traditional Yee＇s scheme： it is a multisymplectic scheme and keeps geometric properties. Moreover, it can be applied on triangular mesh and thus is more adaptive to handle domains with irregular shapes. We have implemented this scheme on a Java platform successfully and our experimental results show that this scheme works well.     

Reply to “Comment on ‘The Imbert-Fedorov shift of paraxial light beams’”

This is a reply to the Comment by Bliokh on our paper that appeared in Opt. Commun. 281 (2008) 3427. After a brief introduction of the representation theory of vector electromagnetic beams advanced in a recent paper, I point out that the Imbert-Fedorov effect is the evidence of the change of the beam parameter Θ and the polarization ellipticity σ caused by the reflection or transmission process in the linear approximation. Then I explain that it is because the linear approximation of the incident beam that we used in our paper had been assumed in previous works that we reproduced their results.     

Numerical solution of 3D Green's function for the dynamic system of anisotropic elasticity

In this Letter, we used homotopy perturbation method to obtain numerical solution of the 3D Green's function for the dynamic system of anisotropic elasticity. Application of homotopy perturbation method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results obtained from convolution of Green's function and data of the Cauchy problem are compared with the exact solution for cubic media. The results reveal that the proposed method is very effective and simple.     

Comment on “The Imbert-Fedorov shift of paraxial light beams”

Here I argue that Liu and Li [B.-Y. Liu, C.-F. Li, Opt. Commun. 281 (2008) 3427] reproduce calculations of the Imbert-Fedorov transverse shift previously made in a number of other works. However, it has recently been shown that these results are not valid for standard uniformly polarized beams. The corrected values of the Imbert-Fedorov shift were derived in papers [K.Y. Bliokh, Y.P. Bliokh, Phys. Rev. Lett. 96 (2006) 073903; Phys. Rev. E 75 (2007) 066609] and confirmed by recent measurements [O. Hosten, P. Kwiat, Science 319 (2008) 787] with a great accuracy.     

Numerical solution of inverse scattering for near-field optics

A novel regularized recursive linearization method is developed for a two-dimensional inverse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium located on a substrate from data accessible through photon scanning tunneling microscopy experiments. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to weak scattering at a low frequency, and each update is obtained by continuation on the wavenumber from solutions of one forward problem and one adjoint problem of the Helmholtz equation.     

An effective method for finding special solutions of nonlinear differential equations with variable coefficients

There are many interesting methods can be utilized to construct special solutions of nonlinear differential equations with constant coefficients. However, most of these methods are not applicable to nonlinear differential equations with variable coefficients. A new method is presented in this Letter, which can be used to find special solutions of nonlinear differential equations with variable coefficients. This method is based on seeking appropriate Bernoulli equation corresponding to the equation studied. Many well-known equations are chosen to illustrate the application of this method.     

Negative refraction of low‐frequency electromagnetic waves

Features of anomalous refraction of low‐frequency electromagnetic waves are investigated in all‐semiconductor nonmagnetic superlattices with a periodic array of two‐dimensional electron gas layers under quantum Hall effect conditions. We show that the structure exhibits negative refraction on the lateral surface for all angles of incidence, in the frequency region which has no limitation from the low‐frequency side, and that for such intriguing properties it requires no resonance frequency and no negative component of the permittivity tensor. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The improved sub-ODE method for a generalized KdV-mKdV equation with nonlinear terms of any order

In this Letter, Li and Wang's sub-ODE method [X.Z. Li, M.L. Wang, Phys. Lett. A 361 (2007) 115] is improved and applied to the generalized KdV-mKdV equation with nonlinear terms of any order. As a result, more travelling wave solutions are obtained including not only all the known solutions found by Li and Wang but also other formal solutions. This improved sub-ODE method can be used for solving other nonlinear partial differential equations with nonlinear terms of any order in mathematical physics.     

On some relations for the inverse blackbody radiation problem

Some relations between the area-temperature distribution of a blackbody and its radiated power spectrum are derived. These relations shall be useful in verifying the adequacy and the accuracy of the various procedures recently developed for the numerical solution of the inverse blackbody radiation problem.