Exact solutions and conservation laws of Zakharov-Kuznetsov modified equal width equation with power law nonlinearity

This paper obtains solutions to the Zakharov-Kuznetsov modified equal width equation with power law nonlinearity. The Lie symmetry approach and the simplest equation method are used to obtain these solutions. Moreover, conservation laws are derived for the underlying equation by employing two approaches: the new conservation theorem and the multiplier method.     

EMDEN-FOWLER TYPE SYSTEM：NOETHER SYMMETRIES AND FIRST INTEGRALS

We classify a generalized coupled singular Emden-Fowler type system +a(t)vn=0,v+b(t)um=0 with respect to the standard first-order Lagrangian according to the Noether point symmetries which it admits.First integrals of the various cases which admit Noether point symmetries are then obtained.This system was discussed in the literature from the view-point of existence and uniqueness of positive solutions.     

Lie–Bäcklund and Noether Symmetries with Applications

New identities relating the Euler–Lagrange, Lie–Bäcklund and Noether operators are obtained. Some important results are shown to be consequences of these fundamental identities. Furthermore, we generalise an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, i.e. a Noether symmetry with zero divergence. We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations. In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilised to construct a Lagrangian for the equation. Several examples are given.     

(Vapor-liquid) equilibria and excess Gibbs free energy functions of (ethanol + glycerol), or (water + glycerol) binary mixtures at several temperatures

The vapor pressures of (ethanol + glycerol) and (water + glycerol) binary mixtures were measured by means of two static devices at temperatures between (273 and 353 (or 363)) K. The data were correlated with the Antoine equation. From these data, excess Gibbs free energy functions (G^E) were calculated for several constant temperatures and fitted to a fourth-order Redlich–Kister equation using the Barker method. The (ethanol + glycerol) binary system exhibits positive deviations in G^E where for the (water + glycerol) mixture, the G^E is negative for all temperatures investigated over the whole composition. Additionally, the NRTL, UNIQUAC and Modified UNIFAC (Do) models have been used for the correlation or prediction of the total pressure.     

Modeling CO2 solubility in Aqueous Methyldiethanolamine Solutions by Perturbed Chain-SAFT Equation of State

CO₂ capture by aqueous alkanolamines treating is one of the prevalent methods to reduce carbon dioxide emissions and to help environmental problems. For realizing more the thermodynamics of the CO₂–MDEA–H₂O, the PC-SAFT equation of state was used to simulate the absorption of carbon dioxide by MDEA (methyldiethanolamine). A correlation for temperature-dependent binary interaction parameter were calculated by excess enthalpy data for aqueous MDEA at low temperatures (lower than 350 K), and then this binary interaction parameter used to predict phase equilibria of ternary aqueous mixtures of MDEA with carbon dioxide. Smith–Missen algorithm and PC-SAFT EOS have been used to determine concentration of species in chemical equilibrium and physical equilibrium, respectively. In addition, for determining parameter sets of MDEA, vapor pressure and saturated liquid density data were used and different and probable association schemes were considered in parameter estimations. Results show 4(2:2, 0:0) association scheme for MDEA and 4(2:2) association scheme for water have better agreement with binary and ternary VLE experimental data.     

Enhancement of the anthracene aqueous solubility by a synergistic effect of alcohols and β-cyclodextrin

The results presented in this paper show that the solubility of anthracene in water increases significantly when β-cyclodextrin and alkanols with carbon numbers from 1 to 4 are added to the solution. The concentrations of the anthracene chosen as a model hydrophobic pollutant were determined using UV–vis spectroscopy. It has been demonstrated that the solubility of anthracene in a 1.1 wt% β-cyclodextrin solution increases by the factor of 45 or 62 when 4 wt% of methanol or ethanol are added, respectively. Obviously, the increase of anthracene solubility is due both to the solvent effect of the alcohols and to the formation of anthracene–β-cyclodextrin (β-CD) “guest–host” complexes. We propose, to use the solubility enhancement based on the β-cyclodextrin/alcohol effect for improving the availability of hydrocarbon pollutants in the biodegradation processes.     

Solutions of Kadomtsev–Petviashvili equation with power law nonlinearity in 1+3 dimensions

This paper studies the solution of the Kadomtsev–Petviasvili equation with power law nonlinearity in 1+3 dimensions. The Lie symmetry approach as well as the extended tanh‐function and G′/G methods are used to carry out the analysis. Subsequently, the soliton solution is obtained for this equation with power law nonlinearity. Both topological as well as non‐topological solitons are obtained for this equation. Copyright © 2010 John Wiley & Sons, Ltd.     

On Huppert’s Conjecture for 3 D 4(q), q ≥ 3

Let G be a finite group and cd(G) be the set of all complex irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G???H × A, where A is an abelian group. In this paper, we verify the conjecture for the family of simple exceptional groups of Lie type ³ D ₄(q), when q?≥?3.     

Travelling Wave Solutions and Conservation Laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equatio

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable $u$ can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions $u$, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach.     

New exact solutions of Bratu Gelfand model in two dimensions using Lie symmetry analysis

We explore new analytical solutions for the two-dimensional nonlinear elliptic Bratu equation. Through the point transformation, the integrable form of Bratu equation was investigated then we obtain the Lie infinitesimals for the new equation. These vectors reduce the integrable equation to solvable ODEs then we use the boundary conditions (BCs) to spin two new exact solutions for Bratu equation in a unit square domain. A three-dimensional plot illustrates some resulting solutions. Comparison with other work has been presented.