A higher‐order accuracy lattice Boltzmann model for the wave equation

A lattice Boltzmann model with higher‐order accuracy for the wave motion is proposed. The new model is based on the technique of the higher‐order moment of equilibrium distribution functions and a series of lattice Boltzmann equations in different time scales. The forms of moments are derived from the binary wave equation by designing the higher‐order dissipation and dispersion terms. The numerical results agree well with classical ones. Copyright © 2008 John Wiley & Sons, Ltd.     

Fresnel coefficients for a layer of NIM

Reflection and transmission of traveling and evanescent waves by a layer of material with a negative index of refraction (NIM) is studied by means of the Fresnel coefficients. We derive their values in the “NIM limit”, and we show that this limit is consistent with the exact solution. It is also indicated that simply substituting the negative values of the relative permittivity and permeability of the NIM material into the exact solution leads to incorrect results for evanescent waves.     

Phonon response of the high‐temperature spin‐density‐wave superconductors

The phonon response of a spin density wave superconductor (SD^W-SC) is calculated. The phonon self energy due to electron-phonon interaction, which involves the electronic density response function, is evaluated explicitly for the coexistent SDW-SC state. It has a square-root singularity at a frequency corresponding to the superconducting gap 2Δ. As a consequence, the spectral density function of the SDW-phonon will show a peak at 2Δ The possibility of observing the 2Δ-peak by Raman scattering is discussed. The relevance of the results to high-temperature superconductors is pointed out. The available Raman data for YBa₂Cu₃O_7-δare compared with the results of the calculation.     

A new auxiliary function method for solving the generalized coupled Hirota–Satsuma KdV system

In this letter, a new auxiliary function method is presented for constructing exact travelling wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the elliptic equation to construct exact travelling wave solutions of nonlinear partial differential equations. More new exact travelling wave solutions are obtained for the generalized coupled Hirota–Satsuma KdV system.     

Quadratic nonlinear torsional hyperelastic waves in isotropic cylinders: Primary analysis of evolution

Nonlinear torsional waves propagating along an infinite isotropic cylinder are considered. The hyperelasticity of the cylinder is described by the Murnaghan potential. An approximate analytical solution of the problem is obtained. The evolution of the initial waveprofile in cylinders made of composite materials reinforced with microscale particles is simulated numerically. It is shown that the approximation used to describe the variation in the amplitude inward the cylinder is quite accurate. Three-dimensional plots of evolution are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 75–85, March 2008.     

A REACTION–DIFFUSION EQUATION MODELING THE INVASION OF THE ARGENTINE ANT POPULATION,LINEPITHEMA HUMILE,AT JASPER RIDGE BIOLOGICAL PRESERVE

Abstract A continuous reaction–diffusion model is developed for the invasive Argentine ant population within a preserve in northern California. The model is a second‐order partial differential equation incorporating a logistic growth term. The dispersal distance traveled during the reproductive process of budding is used to estimate the diffusion coefficient. The model has two homogeneous steady states, one occurring at the propagation front where the Argentine ant population does not yet exist and one occurring where the population has reached carrying capacity. The traveling wave solutions of the model depict the population density for a given time and location. Using current research, parameter values for the model are estimated and a traveling wave solution for the average parameter values is numerically demonstrated.     

Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

A new rational auxiliary equation method and exact solutions of a generalized Zakharov system

A new rational auxiliary equation method for obtaining exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. Its effectiveness is evinced by obtaining exact solutions of a generalized Zakharov system, some of which are new. It is shown that the G^′/G and the generalized projective Ricatti expansion methods are special cases of the auxiliary equation method. Further, due the solutions obtained, four other new and practicable rational methods are deduced.