Finite difference approximations for the fractional advection-diffusion equation

Fractional order diffusion equations are viewed as generalizations of classical diffusion equations, treating super-diffusive flow processes. In this Letter, in order to solve the two-sided fractional advection-diffusion equation, the fractional Crank-Nicholson method (FCN) is given, which is based on shifted Grünwald-Letnikov formula. It is shown that this method is unconditionally stable, consistent and convergent. The accuracy with respect to the time step is of order ²(Δt). A numerical example is presented to confirm the conclusions.     

Destruction of the family of steady states in the planar problem of Darcy convection

We consider natural convection of an incompressible fluid in a porous medium described by the planar Darcy equation. For some boundary conditions, Darcy problem may have non-unique solutions in form of a continuous family of steady states. We are interested in the situation when these boundary conditions are violated. The resulting destruction of the family of steady states is studied via computer experiments based on a mimetic finite-difference approach. Convection in a rectangular enclosure is considered under different perturbations of boundary conditions (heat sources, infiltration). Two scenario of the family of equilibria are found: the transformation to a limit cycle and the formation of isolated convective patterns.     

Influence of Non-uniform Temperature Field on Spectra of Fibre Bragg Grating

We simulate the spectrum characteristics of fibre Bragg grating （FBG） with non-uniform temperature using the transmission matrix method, and the results are analysed. It is found that firstly the modulated coefficient of average refractive index is a very important parameter that influences the spectrum characteristic of the fibre Bragg grating, and secondly the spectrum curves are different in different temperature fields at the same parameter. Hence, we can determine the metrical temperature by analysing the spectrum of fibre Bragg grating.     

Optimization of Mechanically Induced Long-Period Fibre Grating Transmission Spectrum

A new double-layer grating template is designed to reduce the out-of-band loss as much as 1.8dB when the loss of LPo3 reaches 10.2 dB. Meanwhile, we propose a method to remove the sidelobes in the transmission spectra by the adjustment of the thickness of pressure plates. The plate-thickness-induced shift of resonant wavelength and the attenuation of loss peak intensity when removing sidelobes can be modified by the fibre distance and contact point on the pressure plates.     

Birefringence of Irregular-Microstructure Fibre

Birefringence of a microstructure fibre with irregular structures is investigated using the finite element method. The birefringence of each sample is statistically analysed, and the relation between the birefringence and the irregular structure of the microstructure fibre is derived. The results indicate that the irregular structure of the symmetrical microstructure fibre result in the birefringence which is more strongly affected by disorder of air holes than the nonuniformity. Moreover, lower birefringence can be obtained with smaller air holes and larger pitch in the fibre structure.     

Voigt lineshape function as a solution of the parabolic partial differential equation

The goal of this paper is to show that the Voigt function may be found as a solution of a parabolic partial differential equation, like the heat conduction equation or other diffusion equations. A square of the Gaussian half-width of the Voigt function plays the role of ‘time’ and initial conditions are determined by a Lorentz function. Some questions concerning the practical application of the numerical grid methods for the calculation of the Voigt function are discussed. It is shown, that in some cases the offered calculation algorithm can be both faster and more accurate than other known algorithms.     

Fabrication of Pure Silica Core Multimode Ultraviolet Optical Fibre Probes by Tube Etching

As a light wave-guide component for transmitting ultraviolet （UV） laser pulses, pure silica core UV fibre probes have attracted a great deal of attention in the near-field optical data storage and bio-medical studies. We fabricate UV fibre probes with tips in dimension of about 2-5μm and taper angle 16° by the tube etching method, using 40% HF acid as etching solution and xylene as overlayer. Probes produced have curvy configuration with smooth surface. The yield of fine probes is rather high and etching operation greatly simplified. With higher damage threshold, pure silica core multimode UV fibre probes can be coupled into more laser power. In addition, using UV light reduces the cutoff wavelength of the fibre probes, which is in favour of increasing the transmission efficiency of the probe. Furthermore, the larger tip dimension helps to enhance the light throughput either. The advances of fabrication technique of UV optical fibre probe may further support the studies of UV light data storage, pulsed laser biosurgery and UV photolithography.     

Design of Microstructured Optical Fibres with Elliptical Air Holes for Properties： Single-Polarization Single-Mode and Nearly Zero Ultra-Flattened Dispersion

By using the complex finite element method （FEM） under perfectly matched layer （PML） boundary conditions, dispersion properties of microstructured optical fibres （MOFs） with elliptical air holes are analysed by changing the pitch and sizes of air holes belonging to the inner three rings. Meanwhile, the confinement loss of the fundamental mode is engineered to achieve the single-polarization single-mode transmission. Based on this analysis, a novel design of MOFs for properties of the single-polarization single-mode and the nearly zero ultraflattened dispersion between lpskm^-1 nm^-1 in the wavelength range of 1.2-1.6μm is presented for the first time.     

Theoretical Design of Single-Polarization Single-Mode Microstructured Polymer Optical Fibres

Using the full-vector plane-wave expansion method, a kind of PMMA-based polarization-maintaining microstructured optical fibre （PM-mPOF） is theoretically studied. Dependence of the cutoff wavelengths of the two orthogonal polarization states （polarized along the two principal axes of PM-mPOF） on the structure parameters of the fibre is investigated in detail A single-polarization single-mode （SPSM） PM-mPOF working in the visible region is designed and optimized with the result of the maximum SPSM bandwidth of 140nm.     

A Maple Package to Compute Lie Symmetry Groups and Symmetry Reductions of （1＋1）-Dimensional Nonlinear Systems

Armed with the computer algebra system Maple, using a direct algebraic substitution method, we obtain Lie point symmetries, Lie symmetry groups and the corresponding symmetry reductions of one component nonlinear integrable and nonintegrable equations only by clicking the ‘Enter＇ key. Abundant （1＋1）-dimensional nonlinear mathematical physical systems are analysed effectively by using a Maple package LieSYMGRP proposed by us.     

Exact and numerical solutions of generalized Drinfeld-Sokolov equations

In this Letter, we consider a system of generalized Drinfeld-Sokolov (gDS) equations which models one-dimensional nonlinear wave processes in two-component media. We find some exact solutions of gDS by using tanh function method and we also obtain a numerical solution by using the Adomian's Decomposition Method (ADM).     

Initial value problem of the Whitham equations for the Camassa-Holm equation

We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham solution matches the Burgers solution, which exists outside the cusp.     

Propagation of Single-Mode Fibre Laser Beams through an Optical ABCD System with Circular Aperture at the Fibre Output End

Based on the expansion expression of the fundamental mode of a single-mode fibre in terms of Laguerr-Gauss modes, the propagation of a beam of a weakly guiding fibre laser through an opticM ABCD system with a circular aperture at the fibre end is studied. The results show that there is much difference between the propagation of the laser beam described by the expansion expression and by the Gaussian mode approximation. The depth of focus of the laser beam is longer than that of the Gaussian modes.     

Mode Exciting Properties of Photonic Crystal Fibres with the Optical Field Incident from a Single Mode Fibre

We numerically investigate the mode exciting properties of photonic crystal fibres by using the beam propagation method when the optical field is input from a traditional single mode fibre. The results show that both the excited mode spectrum and the coupling-efficiency of each excited mode depend on the normalized pitch А/λ and the normalized hole-size А/λ. Furthermore, we obtain the boundary profile of the optimizing coupling-efficiency for the excited fundamental mode： the boundary （А/λ）＊ is linear to the boundary （А/λ）. All of these will pave the way for smoothing applications of photonic-crystal fibres, such as splicing and designing photonic-crystal-fibre functional devices.     

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).     

Painleve Property and New Analytic Solutions for a Variable-Coefficient Kadomtsev--Petviashvili Equation with Symbolic Computation

A variable-coemcient Kadomtsev-Petviashvili equation is investigated. The Painlev6 analysis leads to its explicit Painlevd-integrable conditions. An auto-Backlund transformation and the bilinear form are presented via the truncated Painlev6 expansion and symbolic computation. Several families of new analytic solutions arepresented, including the soliton-like and periodic solutions.     

A Field Method for Integrating Equations of Motion of Nonlinear Mechanico-Electrical Coupling Dynamical Systems

We deal with the generalization of the field method to weakly non-linear mechanico-electrical coupling systems. The field co-ordinates and field momenta approaches are combined with the method of multiple time scales in order to obtain the amplitudes and phase of oscillations in the first approximation. An example in mechanico-electrical coupling systems is given to illustrate this method.     

Linearization of Systems of Nonlinear Diffusion Equations

We investigate the linearization of systems of n-component nonlinear diffusion equations; such systems have physical applications in soil science, mathematical biology and invariant curve flows. Equivalence transformations of their auxiliary systems are used to identify the systems that can be linearized. We also provide several examples of systems with two-component equations, and show how to linearize them by nonlocal mappings.     

Lie Algebra and Lie Super Algebra for Integrable Couplings of C-KdV Hierarchy

Based on the constructed Lie algebra and Lie super algebra, the integrable couplings and super-integrable couplings of the C-KdV hierarchy are obtained respectively. Furthermore, its super-Hamiltonian structures are presented by using super-trace identity.     

Integrable aspects and applications of a generalized inhomogeneous N-coupled nonlinear Schrödinger system in plasmas and optical fibers via symbolic computation

For describing the general behavior of N fields propagating in inhomogeneous plasmas and optical fibers, a generalized N-coupled nonlinear Schrödinger system is investigated with symbolic computation in this Letter. When the coefficient functions obey the Painlevé-integrable conditions, the (N+1)×(N+1) nonisospectral Lax pair associated with such a model is derived by means of the Ablowitz-Kaup-Newell-Segur formalism. Furthermore, the Darboux transformation is constructed so that it becomes exercisable to generate the multi-soliton solutions in a recursive manner. Through the graphical analysis of some exact analytic one- and two-soliton solutions, our discussions are focused on the envelope soliton excitation in time-dependent inhomogeneous plasmas and the optical pulse propagation with the constant (or distance-related) fiber gain/loss and phase modulation.