Volume 65, Issue 4

In this paper, we apply a modified version of the variational iteration method (MVIM) for solving Burgers’ and coupled Burgers’ equations. The proposed modification is made by introducing He’s polynomials in the correction functional of the variational iteration method (VIM). The use of Lagrange multiplier coupled with He’s polynomials are the clear advantages of this technique over the decomposition method.

An efficient numerical method is developed for solving nonlinear wave equations by studying the propagation and stability properties of solitary waves (solitons) of the regularized long wave (RLW) equation in one space dimension using a combination of leap frog for time dependence and a pseudospectral (Fourier transform) treatment of the space dependence. Our schemes follow very accurately these solutions, which are given by simple closed formulas. Studying the interaction of two such solitons and three solitary waves interaction for the RLW equation. Our implementation employs the fast Fourier transform (FFT) algorithm.

In this paper, we apply the modified variational iteration method (mVIM) for solving integrodifferential equations and coupled systems of integro-differential equations. The proposed modification is made by the elegant coupling of He’s polynomials and the correction functional of variational iteration method. The proposed mVIM is applied without any discretization, transformation or restrictive assumptions and is free from round off errors and calculation of the so-called Adomian’s polynomials.

An analytic technique, the homotopy analysis method (HAM), is applied to obtain the kink solution of the Sharma-Tasso-Olver equation. The homotopy analysis method is one of the analytic methods and provides us with a new way to obtain series solutions of such problems. HAM contains the auxiliary parameter ħwhich gives us a simple way to adjust and control the convergence region of series solution. “Due to this reason, it seems reasonable to rename ħthe convergence-control parameter” [1].

Under investigation in this paper, with symbolic computation, is the Sasa-Satsuma (SS) equation which can describe the propagation of ultra short pulses in optical fiber communications. By virtue of the Ablowitz-Kaup-Newell-Segur procedure, the Lax pair for the SS equation is directly established. Based on such a Lax pair, a Bäcklund transformation is constructed, through which the explicit onesoliton solution is derived.Meanwhile, an infinite number of conservation laws is provided to indicate the integrability of the SS equation in the Liouville sense. To further understand the stability of the one-soliton solution, we employ the split-step Fourier method to simulate the propagation of the soliton pulses under the finite initial perturbations. In addition, the interaction of two adjacent pulses with different separation distances is investigated through numerical simulation. Analytic and numerical results discussed in this paper are expected to be applied to the description of the optical pulse propagation.

Using the homotopy perturbation method, a class of nonlinear generalized Ginzburg-Landau-Higgs systems (GGLH) is considered. Firstly, by introducing a homotopic transformation, the nonlinear problem is changed into a system of linear equations. Secondly, by selecting a suitable initial approximation, the approximate solution with arbitrary degree accuracy to the generalized Ginzburg- Landau-Higgs system is derived. Finally, another type of homotopic transformation to the generalized Ginzburg-Landau-Higgs system reported in previous literature is briefly discussed.

Efficient numerical methods for solving nonlinear wave equations and studying the propagation and stability properties of their solitary waves (solitons) are applied to a Boussinesq type equation in one space dimension. These methods use a pseudospectral (Fourier transform) treatment of the space dependence, together with (a) finite differences, or (b) a fourth-order Runge-Kutta scheme (RK4), for the time evolution. Our schemes follow very accurately single solitons, which are given by simple closed formulas and are known to be stable for all allowed velocities. However, as a parameter of the problem tends to the critical value b = 0 . 5, where the velocity of the exact soliton vanishes, our solutions destabilize due to numerical errors, producing two small solitons in the place of the exact one. On the other hand, when we study the interaction of two such solitons, starting far apart from each other, we find in the b 1 , b 2 parameter plane a curve beyond which the solution becomes unstable by exponential blow-up of the amplitudes, independently of our space and time discretization. We claim that this is due to a dynamical resonance rather than the accumulation of numerical errors, in agreement with theoretical predictions. Our implementation relies on the fast Fourier transform (FFT) algorithm and no major differences are observed, when either scheme (a) or (b) is used for the evolution of time.

In the present investigation, we have studied ion-acoustic solitary waves in a plasma consisting of warm positive and negative ions and nonisothermal electron distribution. We have used reductive perturbation theory (RPT) and derived a dispersion relation which supports only two ion-acoustic modes, viz. slow and fast. The expression for phase velocities of these modes is observed to be a function of parameters like nonisothermality, charge and mass ratio, and relative temperature of ions. A modified Korteweg-de Vries (KdV) equation with a (1+1/2) nonlinearity, also known as Schamel-mKdV model, is derived. RPT is further extended to include the contribution of higher-order terms. The results of numerical computation for such contributions are shown in the form of graphs in different parameter regimes for both, slow and fast ion-acoustic solitary waves having several interesting features. For the departure from the isothermally distributed electrons, a generalized KdV equation is derived and solved. It is observed that both rarefactive and compressive solitons exist for the isothermal case. However, nonisothermality supports only the compressive type of solitons in the given parameter regime.

The transmission properties of one-dimensional photonic crystals constituted by a periodic repetition of positive-index layers and epsilon-negative layers are studied theoretically. This structure shows some interesting properties including a wide gap in the low frequency range for small period number and a comb-like transmission band in the gap. The properties of the comb-like transmission band are sensitive to the period number of the structure. In contrast to the zero-ngap and the zero- φ eff gap, the transmission properties are dependent on the structure parameters. A general method to decide the position of gap and transmission band in this kind of structure is also presented.

The tri critical point (TCP), where the second-order transition transforms to first order has been located in nO.m homologous series. The order parameter has been estimated from the birefringence δn, from the refractive indices and from birefringence data available in literature and from those obtained at our laboratory on a number of nO.m compounds. The compounds in the nO.m series exhibit both second and first-order nematic/smectic-A (NS A ) transition depending on the McMillan ratio (T NA /T IN ) which in turn depends on the nematic and smectic-A thermal ranges. The data presented are compared with the body of the data available on this homologous series obtained with other techniques.

The determination of texture effects in nickel-cobalt (Ni-Co) films with different thickness, which were obtained by electrodeposition, has been investigated by the measurement of hysteresis loops at different angles. Easy-axis distribution measurements were performed as a function of the squareness M p(β ) and the correlations were established among the different thicknesses. The composition of Ni- Co films was determined by energy dispersive X-ray spectroscopy. The structural analysis made by X-ray diffraction revealed that all films have a polycrystalline face-centered cubic structure but their texture degrees vary depending on the film thickness. The determination of the easy-axis orientation in 2-D films from the M p (β ) obtained by the hystersis loops was studied using Fourier series analyses. The coefficient A 0 have a value of less then unity while A 2 is inversely proportional to the width of the distribution function which may cause the change in the texture preferential orientations. Therefore, the differences observed in the magnetic easy-axis distributions were attributed to the changes in texture orientations caused by the compositional differences at different thicknesses of the polycrystalline films.

Electron paramagnetic resonance (EPR) studies are carried out on vanadyl (VO 2+ ) ions in diammonium tricadmium tetrakis (sulfate) pentahydrate single crystals at room temperature. The EPR spectra of a single crystal exhibit resonance signals characteristic to VO 2+ ions. The analysis of EPR spectra indicates that the VO 2+ ions in single crystals show two magnetically inequivalent VO 2+ sites in distinct orientations occupying substitutional positions in the lattice and showing very high angular dependence. They form in octahedral coordination with tetragonal compression with C 4 v symmetry. The spin Hamiltonian parameters are determined, and these parameters have been used to estimate the bonding coefficients of the VO 2+ ion in a diammonium tricadmium tetrakis (sulfate) pentahydrate lattice. The parallel and perpendicular components of axially symmetric g and hyperfine (A) tensors are evaluated and the results are discussed and compared with previous reports.

We apply the Weiss-Tabor-Carnevale (WTC) Painlev´e test to members of a sequence of higher-order shallow-water type equations. We obtain the result that the equations considered are non-integrable, although compatibility conditions at real resonances are satisfied. We also construct travelling-wave solutions for these and related equations.