Painlevé analysis, Lie symmetries and exact solutions for (2+1)-dimensional variable coefficients Broer-Kaup equations

A (2+1) dimensional Broer-Kaup system which is obtained from the constraints of the KP equation is of importance in mathematical physics field. In this paper, the Painlevé analysis of (2+1)-variable coefficients Broer-Kaup (VCBK) equation is performed by the Weiss-Kruskal approach to check the Painlevé property. Similarity reductions of the VCBK equation to one-dimensional partial differential equations including Burger’s equation are investigated by the Lie classical method. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.     

Convergence Analysis of Weighted Difference Approximations on Piecewise Uniform Grids to a Class of Singularly Perturbed Functional Differential Equations

We consider the numerical approximation of a singularly perturbed time delayed convection diffusion problem on a rectangular domain. Assuming that the coefficients of the differential equation be smooth, we construct and analyze a higher order accurate finite difference method that converges uniformly with respect to the singular perturbation parameter. The method presented is a combination of the central difference spatial discretization on a Shishkin mesh and a weighted difference time discretization on a uniform mesh. A?priori explicit bounds on the solution of the problem are established. These bounds on the solution and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. It is shown that the proposed method is $L_{2}^{h}$ -stable. The analysis done permits its extension to the case of adaptive meshes which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practical satisfies the theoretical predictions.     

Synthesis of acetylenes via dehydrobromination using solid anhydrous potassium phosphate as the base under phase-transfer conditions

Phase-transfer catalyzed preparation of acetylenes from the corresponding vicinal dibromo compounds via double dehydrobromination using the mild solid base, anhydrous potassium phosphate, under very mild conditions is reported.     

Stereoselective synthesis of oxylipin-3: [(6R,9R,10R,7E)-6,9,10-trihydroxyoctadec-7-enoic acid]

The enantioselective synthesis of oxylipin-3 has been carried out. The synthetic strategy was started from commercially available n-decanal. The chiral source was based on Sharpless asymmetric dihydroxylation and CBS reduction.     

Direct hapten-linked competitive inhibition enzyme-linked immunosorbent assay (CIELISA) for the detection of O-pinacolyl methylphosphonic acid

Immunoassay detection of O-pinacolyl methylphosphonic acid (PMPA) employing direct coating of N-2-aminoethyl-O-pinacolyl methylphosphonate (hapten B) on microtiter plates is reported. Coating was achieved by covalently linking hapten B to a glutaraldehyde (GA) polymer network directly bound to the polystyrene (PS) surface of a standard 96-well microtiter plate. 4-(2-(O-Pinacolylmethylphosphoryl amino)ethyl amino)-4-oxobutanoic acid (hapten A)-ovalbumin (OVA) conjugate served as the coating antigen for comparison with direct hapten B-coated plates in the CIELISA format. The developed assay employing direct hapten B coated plates demonstrated enhanced sensitivity with the IC(50) value for PMPA being 0.027 μg mL(-1). The assay could detect PMPA even at the concentration of 0.006 μg mL(-1). The mean recovery of standard PMPA (spiked in water) was found to be 83.7%.     

Luminescence Properties of Dual Valence Eu Doped Nano-crystalline BaF2 Embedded Glass-ceramics and Observation of Eu2+ → Eu3+ Energy Transfer

Europium doped glass-ceramics containing BaF₂ nano-crystals have been prepared by using the controlled crystallization of melt-quenched glasses. X-ray diffraction and transmission electron microscopy have confirmed the presence of cubic BaF₂ nano-crystalline phase in glass matrix in the ceramized samples. Incorporation of rare earth ions into the formed crystalline phase having low phonon energy of 346 cm⁻¹ has been demonstrated from the emission spectra of Eu³⁺ ions showing the transitions from upper excitation states ⁵D_J (J = 1, 2, and 3) to ground states for the glass-ceramics samples. The presence of divalent europium ions in glass and glass-ceramics samples is confirmed from the dominant blue emission corresponding to its 5d-4f transition under an excitation of 300 nm. Increase in the reduction of trivalent europium (Eu³⁺) ions to divalent (Eu²⁺) with the extent of ceramization is explained by charge compensation model based on substitution defect mechanisms. Further, the phenomenon of energy transfer from Eu²⁺ to Eu³⁺ ion by radiative trapping or re-absorption is evidenced which increases with the degree of ceramization. For the first time, the reduction of Eu³⁺ to Eu²⁺ under normal air atmospheric condition has been observed in a BaF₂ containing oxyfluoride glass-ceramics system.     

Self focusing of laser beams in a degenerate nonparabolic semiconductor: Kinetic approach

Using rigorous kinetic treatment the steady state self-focusing of laser beams in a degenerate nonparabolic semiconductor (e.g. n-InSb) has been investigated beyond the perturbation limit. The nonlinearity arises due to the energy dependence of the electron mass in the conduction band. The energy loss of electron energy is assumed to be due to polar optical phonons and momentum transfer due to ionised impurity scattering/polar optical phonon scattering. It is found that nonlinearity in the dielectric constant or the self-focusing is more pronounced in the case of ionised impurity case as compared to polar optical phonon case. The effect of absorption is to suppress the self-focusing but the latter is also enhanced by degeneracy.     

Bis(indenyl)titanium(IV) and zirconium(IV) complexes of monofunctional bidentate salicylidimines

Dichlorobis(indenyl)-titanium(IV) and -zirconium(IV), (C₉H₇)₂TiCl₂ and (C₉H₇)₂ZrCl₂, react with bidentate Schiff bases such as salicylidene aniline, salicylidene-o-toluidine, salicylidene-m-toluidine and salicylidene-p-toluidine in a 1:1 molar ratio in refluxing tetrahydrofuran in the presence of triethylamine to yield complexes of the type (C₉H₇)₂Ti(SB)Cl and (C₉H₇)₂Zr(SB)Cl, respectively where SB is the anion of the corresponding Schiff base, SBH. The new derivatives have been characterised on the basis of their elemental analyses, conductance measurements and spectral (IR, ¹H NMR and electronic) studies.