A lattice Bhatnagar--Gross--Krook model for a class of the generalized Burgers equations

A new lattice Bhatnagar--Gross--Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods. Satisfactory results are obtained by the numerical simulations.     

Novel methods for finding general forms of new multi-soliton solutions to (1+1)-dimensional KdV equation and (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation

In this paper, two novel methods used to solve (1+1) and (2+1)-dimensional completely integrable equations are proposed. The methods are applied to handle the KdV and Kadomtsev–Petviashvili (KP) equations with variable coefficients, and the general forms of new multi-soliton solutions are formally obtained, respectively. In addition, the new multi-soliton solution is suitable to two different type KP equations. Comparing with the Hirota’s method, the results show that new methods are straightforward handling the KdV and KP equations without conjecturing the transformation and good in dealing the equations with variable coefficients.     

Solutions to Class of Linear and Nonlinear Fractional Differential Equations

In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the(3+1)-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.     

Scaling behavior of the time-fractional equations for molecular-beam epitaxy growth: scaling analysis versus numerical stimulations

The scaling behavior of the time-fractional molecular-beam epitaxy (TFMBE) equations in 1+1 dimensions is investigated by numerical simulations and scaling analysis, respectively. The growth equations studied include the time-fractional linear molecular-beam epitaxy (TFLMBE) and the time-fractional Lai-Das Sarma-Villain (TFLDV). Growth exponents are obtained using the two methods. The analytical results are consistent with the corresponding numerical solutions based on Caputo-type fractional derivative.     

Bifurcation methods of dynamical systems for handling nonlinear wave equations

By using the bifurcation theory and methods of dynamical systems to construct the exact travelling wave solutions for nonlinear wave equations, some new soliton solutions, kink (anti-kink) solutions and periodic solutions with double period are obtained.      

广义Birkhoff方程的积分方法

场方法和最终乘子法是求解运动微分方程的基本方法.本文将这两种方法应用于广义Birkhoff系统,求出了场方法的基本偏微分方程和该方程的完全积分;根据Jacobi最终乘子定理求出了广义Birkhoff方程的解.并举例说明结果的应用.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions are performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

Explicit Solutions for Generalized (2+1)-Dimensional NonlinearZakharov-Kuznetsov Equation

The exact solutions of the generalized (2+1)-dimensional nonlinearZakharov-Kuznetsov (Z-K) equation are explored by the method of the improved generalized auxiliary differential equation. Many explicit analytic solutions of the Z-K equation are obtained. The methods used to solve the Z-K equation can be employed in further work to establish new solutions for other nonlinear partial differential equations.     

New Exact Solutions of Broer-Kaup Equations

By a known transformation, (2 1)-dimensional Brioer-Kaup equations are turned to a single equation.The classical Lie symmetry analysis and similarity reductions axe performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.     

Solvability of a class of PT-symmetric non-Hermitian Hamiltonians:Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrdinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl(2) hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl(2) Lie algebra. It is found that the results of the two methods are the same.     

Analytical solutions for the unsteady MHD rotating flow over a rotating sphere near the equator

In this paper we investigate the three-dimensional magnetohydrodynamic (MHD) rotating flow of a viscous fluid over a rotating sphere near the equator. The Navier-Stokes equations in spherical polar coordinates are reduced to a coupled system of nonlinear partial differential equations. Self-similar solutions are obtained for the steady state system, resulting from a coupled system of nonlinear ordinary differential equations. Analytical solutions are obtained and are used to study the effects of the magnetic field and the suction/injection parameter on the flow characteristics. The analytical solutions agree well with the numerical solutions of Chamkha et al. [31]. Moreover, the obtained analytical solutions for the steady state are used to obtain the unsteady state results. Furthermore, for various values of the temporal variable, we obtain analytical solutions for the flow field and present through figures.     

Application of the Generalized Differential Quadrature Method in Solving Burgers'Equations

The aim of this paper is to obtain numerical solutions of the one-dimensional, two-dimensional and coupled Burgers' equations through the generalized differential quadrature method (GDQM). The polynomial-based differential quadrature (PDQ) method is employed and the obtained system of ordinary differential equations is solved via the total variation diminishing Runge-Kutta (TVD-RK) method. The numerical solutions are satisfactorily coincident with
the exact solutions. The method can compete against the methods applied in the literature.     

Quasinormal modes of a Schwarzschild black hole immersed in an electromagnetic universe

We study the quasinormal modes(QNMs) of a Schwarzschild black hole immersed in an electromagnetic(EM) universe. The immersed Schwarzschild black hole(ISBH) originates from the metric of colliding EM waves with double polarization [Class. Quantum Grav. 12, 3013(1995)]. The perturbation equations of the scalar fields for the ISBH geometry are written in the form of separable equations. We show that these equations can be transformed to the confluent Heun's equations, for which we are able to use known techniques to perform analytical quasinormal(QNM) analysis of the solutions. Furthermore, we employ numerical methods(Mashhoon and 6~(th)-order Wentzel-Kramers-Brillouin(WKB)) to derive the QNMs. The results obtained are discussed and depicted with the appropriate plots.     

An Analytical Self-consistent Solution for the Free Energy of a 1-D Chain of Atoms Including Anharmonic Effects

The self-consistent method of lattice dynamics (SCLD) is used to obtain an analytical solution for the free energy of a periodic, one-dimensional, mono-atomic chain accounting for fourth-order anharmonic effects. For nearest-neighbor interactions, a closed-form analytical solution is obtained. In the case where more distant interactions are considered, a system of coupled nonlinear algebraic equations is obtained (as in the standard SCLD method) however with the number of equations dramatically reduced. The analytical SCLD solutions are compared with a numerical evaluation of the exact solution for simple cases and with molecular dynamics simulation results for a large system. The advantages of SCLD over methods based on the harmonic approximation are discussed as well as some limitations of the approach.     

Linearization procedure for cylindrically and axially symmetric space-times

An investigation of the vacuum Einstein gravitational field equations for cylindrically and axially symmetric space-times is presented which leads to an equivalent differential system involving a simple nonlinearity only. The case when this equivalent system is linear is analyzed in detail and two methods for generating solutions of the Einstein vacuum equations are set up. As a result, in the axially symmetric case the linearity of the equivalent system characterizes completely the Kramer-Neugebauer transforms of Papapetrou line elements. Accordingly, Weyl solutions are shown to generate exhaustively both Lewis and van Stockum solutions. Analogous results are obtained also in the cylindrically symmetric case.     

Travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms

Applying the general projective Riccati equations method, we consider the exact travelling wave solutions for generalized symmetric regularized long-wave equations with high-order nonlinear terms using symbolic computation.From our results, we not only can successfully recover some previously known travelling wave solutions found by using various tanh methods, but also can obtain some new formal solutions. The solutions obtained include kink-shaped solitons, bell-shaped solitons, singular solitons and periodic solutions.     

Classical solutions to topologically massive Yang-Mills theory

Classical solutions to (2 + 1)-dimensional Yang-Mills theory in the presence of the Chern-Simons invariant are considered. The SO(3)-invariant solutions to the Euclidean field equations are complex, whereas the equations in Minkowski space-time possess real SO(2, 1)-invariant solutions. The field equations for time independent axially symmetric vector potentials are derived and some solutions are obtained. The behavior of general Euclidean spacetime solutions is discussed. It is also shown that, because of the gauge dependence of the Chern-Simons invariant, the reduced field equations cannot be uniquely obtained from the reduced action. Applications of the results to the infrared structure of finite temperature QCD are discussed; in particular, it is argued that the Chern-Simons invariant cannot be consistently incorporated as a gauge-invariant magnetic mass term in a three-dimensional effective long distance theory at high temperatures.     

Implementation of three reliable methods for finding the exact solutions of (2 + 1) dimensional generalized fractional evolution equations

In this study, we have implemented the three methods namely extended \((G^{\prime}/G)\)-expansion, extended \((1/G^{\prime})\)-expansion and \((G^{\prime}/G,\,\,1/G)\)-expansion methods to determine exact solutions for the (2 + 1) dimensional generalized time–space fractional differential equations. We use Conformable fractional derivative and its properties in this research to convert fractional differential equations to ordinary differential equations with integer order. By using above mentioned methods, three types of traveling wave solutions are successfully obtained which have been expressed by the hyperbolic, trigonometric, and rational function solutions. The considered methods and transformation techniques are efficient and consistent for solving nonlinear time and space-fractional differential equations.     

Modeling of Dark Solitons for Nonlinear Longitudinal Wave Equation in a Magneto-Electro-Elastic Circular Rod

In this paper, sub equation and expansion methods are proposed to construct exact solutions of a nonlinear longitudinal wave equation (LWE) in a magneto-electro-elastic circular rod. The proposed methods have been used to construct hyperbolic, rational, dark soliton and trigonometric solutions of the LWE in the magneto-electro-elastic circular rod. Arbitrary values are given to the parameters in the solutions obtained. 3D, 2D and contour graphs are presented with the help of a computer package program. Solutions attained by symbolic calculations revealed that these methods are effective, reliable and simple mathematical tool for finding solutions of nonlinear evolution equations arising in physics and nonlinear dynamics.