On solving the nonlinear Biswas-Milovic equation with dual-power law nonlinearity using the extended tanh-function method

In this article, we apply the extended tanh-function method to find the exact traveling wave solutions of the nonlinear Biswas-Milovic equation (BME), which describes the propagation of solitons through optical fibers for trans-continental and trans-oceanic distances. This equation is a generalized version of the nonlinear Schrödinger equation with dual-power law nonlinearity. With the aid of computer algebraic system Maple, both constant and time-dependent coefficients of BME are discussed. Comparison between our new results and the well-known results is given. The given method in this article is straightforward, concise and can be applied to other nonlinear partial differential equations (PDEs) in mathematical physics

Exact Traveling Wave Solutions of Nonlinear PDEs in Mathematical Physics Using the Modified Simple Equation Method

In this article, we apply the modified simple equation method to find the exact solutions with parameters of the (1+1)-dimensional nonlinear Burgers-Huxley equation, the (2+1) dimensional cubic nonlinear Klein-Gordon equation and the (2+1)-dimensional nonlinear Kadomtsev- Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The new exact solutions of these three equations are obtained. When these parameters are given special values, the solitary solutions are obtained

Genetic variations between two ecotypes of Egyptian clover by inter-simple sequence repeat (ISSR) techniques

The inter-simple sequence repeat (ISSR) markers have been used in order to determine genetic variation and relationship between two clover ecotypes. Ten (10) primers for ISSR were used in this study but only six were successful in generating reproducible and reliable amplicons for different types of the Egyptian clover. The results reveal the polymorphism level by ISSR primers. HB10 ISSR-primer was higher than the rest of the ISSR primers in polymorphic 100%. The Fahl monocut ecotype had 29 present bands, 3 absent bands in total of 32 bands; among those there were two unique bands. The multicut ecotype were given different pattern of bands, Gemmiza1 (21 present and 11 absent), Giza6 (21 present and 11 absent) and Serw1; (23 present and 9 absent). There were three unique bands appearance in the two ecotypes. Fahl was given two with HB11 and HB13; the Serw1 multicut cultivar had one unique bands with HB08. Similarity indices among the four Egyptian clover cultivars based on ISSR analysis was estimated and the highest value appeared between Fahl and Gemmiza1 as well as Giza6 and Serw1 followed by Fahl and Serw1. The lowest similarity value appeared between Gemmiza1 and Serw1 followed by Gemmiza1 and Giza6.Key words: Egyptian clover, molecular marker, Fahl, Gemmiza1, Giza6, Serw1

Melting and Solidification Study of As-Deposited and Recrystallized Bi Thin Films

Melting and solidification of as-deposited and recrystallized Bi crystallites, deposited on highly oriented 002-graphite at 423 K, were studied using reflection high-energy electron diffraction (RHEED). Films with mean thickness between 1.5 and 33 ML (monolayers) were studied. Ex situ atomic force microscopy was used to study the morphology and the size distribution of the formed nanocrystals. The as-deposited films grew in the form of three-dimensional crystallites with different shapes and sizes, while those recrystallized from the melt were formed in nearly similar shapes but different sizes. The change in the RHEED pattern with temperature was used to probe the melting and solidification of the crystallites. Melting started at temperatures below the bulk melting point of Bi, T0=544.5 K, and extended over a temperature range that depended on the size distribution of the crystallites. The as-deposited 1.5 ML film started to melt at T0-50 K and melted completely at T0-20 K. For films with higher coverage, the size distribution was observed to spread over a wider range with a larger mean value, resulting in a shift in the melting temperature range towards higher temperatures. Due to the shift in size distribution to higher values upon recrystallization, the recrystallized Bi crystallites showed a melting temperature range higher than that of the as-deposited crystallites. For the investigated conditions, all films were completely melted below or at T 0 of Bi. The characteristic film melting point, defined as the temperature at which the film melting rate with temperature is the fastest, showed a linear dependence on the reciprocal of the average crystallite radius, consistent with theoretical models. Of these models, the surface-phonon instability model best fits the obtained results. During solidification, the Bi films showed high amount of supercooling relative to T0 of Bi. The amount of liquid supercooling was found to decrease linearly with the reciprocal of the average crystallite size. © 2006 American Institute of Physics. [DOI: 10.1063/1.2208551

Condensation on (002) Graphite of Liquid Bismuth Far Below Its Bulk Melting Point

Condensation of thermally evaporated Bi on (002) graphite, at temperatures of 300-523K, was studied using in situ reflection high-energy electron diffraction (RHEED) and room temperature ex situ atomic force microscopy (AFM). For deposition at temperatures below 415±5K, transmission RHEED patterns of Bi appeared at an average thickness of ∼0.5 monolayer (ML). AFM images showed that the film consisted of crystallites in the shape of triangular step pyramids with step heights corresponding to single and double Bi layers in the [111] direction. This morphology indicates crystallization from the vapor. For deposition at higher temperatures, diffuse RHEED patterns appeared independent of the deposited thickness. When these films were cooled, clear transmission patterns of crystalline Bi appeared. After cooling to near room temperature, the melting and solidification behaviors of these films were investigated with RHEED. Upon subsequent heating, the topmost layers of the probed Bi crystallites started to lose long-range order at ∼10-15K below the Bi bulk melting point, T0=544.52K. When crystallized from the melt, supercooling by ∼125K below T0 was observed. These results indicate that Bi condensed on graphite in the form of supercooled liquid droplets when the graphite temperature was above 419K (T0-125). Below that temperature, Bi condensed in the solid phase. Bi films crystallized by cooling the liquid had crystal morphologies that depended on the degree of liquid supercooling. © 2005 The American Physical Society

Generalized and Improved (G'/G)-Expansion Method for Nonlinear Evolution Equations

A generalized and improved (G'/G)-expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov- Benjamin-Bona-Mahony �ZKBBM� equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering

Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

The spectral function Θ(t)=∑i=1∞exp(−tλj), where {λj}j=1∞ are the eigenvalues of the negative Laplace-Beltrami operator −Δ, is studied for a compact Riemannian manifold Ω of dimension k with a smooth boundary ∂Ω, where a finite number of piecewise impedance boundary conditions (∂∂ni+γi)u=0 on the parts ∂Ωi(i=1,…,m) of the boundary ∂Ω can be considered, such that ∂Ω=∪i=1m∂Ωi, and γi(i=1,…,m) are assumed to be smooth functions which are not strictly positive

An inverse problem for a general annular drum with positive smooth functions in the Robin boundary conditions

The asymptotic expansion of the trace of the heat kernel $\Theta(t) =\sum^\infty_{j=1}$ exp($-t\lambda_j)$ as $t\rightarrow 0^+$ has been derived for a variety of domains, where $\{\lambda_j\}$ are the eigenvalues of the negative Laplace operator $-\Delta = -\sum^2_{i=1}(\frac{\partial}{\partial x^i} )^2$ in the $(x^1, x^2)$-plane. The dependence of $\Theta(t)$ on the connectivity of domains and the boundary conditions is analyzed. Particular attention is given for a general annular drum in $\mathbb{R}^2$ together with Robin boundary conditions, where the coefficients in the boundary conditions are positive smooth functions. Some applications of an ideal gas enclosed in the general annular drum are given

An inverse eigenvalue problem for an arbitrary multiply connected bounded region: an extension to higher dimensions

The basic problem in this paper is that of detemnining the geometry of an arbitrary multiply connected bounded region in R3 together with the mixed boundary conditions, from the complete knowledge of the eigenvalues {λj}j=1∞ for the negative Laplacian, using the asymptotic expansion of the spectral function θ(t)=∑j=1∞exp(−tλj) as t→0