Abstract:
In this paper, the Fisher equation is analysed. One of its travelling wave solution is obtained by comparing it with KdV--Burgers (KdVB) equation. Its amplitude, width and speed are investigated. The instability for the higher order disturbances to the solution of the Fisher equation is also studied.

Abstract:
We present finite difference schemes for Burgers equation and Burgers-Fisher equation. A new version of exact finite difference scheme for Burgers equation and Burgers-Fisher equation is proposed using the solitary wave solution. Then nonstandard finite difference schemes are constructed to solve two equations. Numerical experiments are presented to verify the accuracy and efficiency of such NSFD schemes. 1. Introduction During the last few decades, nonlinear diffusion equation (1) has played an important role in nonlinear physics. Recently, it also began to become important in various other fields of science, for example, biology, chemistry, and economics [1–3]. When , (1) is reduced to the famous Burgers equation (2) This equation is the simplest equation combining both nonlinear propagation effects and diffusive effects [3]. It has been used in many fields especially for describing wave processes in acoustics and hydrodynamics [2]. Researchers have devoted a lot of efforts to studying the solutions of this equation [1–6]. A. van Niekerk and F. D. van Niekerk [4] applied Galerkin methods to the nonlinear Burgers equation and obtained implicit and explicit algorithms using different higher order rational basis functions. Hon and Mao [5] applied the multiquadric as a spatial approximation scheme for solving the nonlinear Burgers equation. Biazar and Aminikhah [6] considered the variational iteration method to solve nonlinear Burgers equation. If we take , (1) becomes the Burgers-Fisher equation (3) Burgers-Fisher equation is very important in fluid dynamic model. There have been extensive studies and applications of this model. A nonstandard finite difference scheme for the Burgers-Fisher equation was given by Mickens and Gumel [7]. In [8], Kaya and El-Sayed constructed a numerical simulation and explicit solutions of the generalized Burgers-Fisher equation. Ismail et al. [9] obtained the approximate solutions for the Burgers-Huxley and Burgers-Fisher equations by using the Adomian decomposition method. Wazwaz [10] presented the tanh method for generalized forms of nonlinear heat conduction and Burgers-Fisher equations. Javidi and Golbabai [11, 12] studied spectral collocation method and spectral domain decomposition method for the solution of the generalized Burgers-Fisher equation. Numerical solution of Burgers-Fisher equation is presented based on the cubic B-spline quasi-interpolation by Zhu and Kang [13]. Kocacoban et al. [14] solved Burgers-Fisher equation by using a different numerical approach that shows rather rapid convergence than other

Abstract:
A method is suggested to obtain the quasi exact solution of the Rabi Hamiltonian. It is conceptually simple and can be easily extended to other systems. The analytical expressions are obtained for eigenstates and eigenvalues in terms of orthogonal polynomials.

Abstract:
Some new exact solutions of the Burgers--Fisher equation and generalized Burgers--Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular solitary wave solutions and some more complex solutions whose figures are given in the article. The result shows that the first integral method is one of the most effective approaches to obtain the solutions of the nonlinear partial differential equations.

Abstract:
Two analytical methods have been developed for constructing approximate solutions to a nonlocal generalization of the 1D Fisher-Kolmogorov-Petrovskii-Piskunov equation. This equation is of special interest in studying the pattern formation in microbiological populations. In the greater part of the paper, we consider in detail a semiclassical approximation method based on the WKB-Maslov theory under the supposition of weak diffusion. The semiclassical asymptotics are sought in a class of trajectory concentrated functions. Such functions are localized in a neighborhood of a point moving in space. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval which can be small in the sense that a pattern has no time to form in this interval. In the final part of the paper, we have constructed asymptotics which are different from the semiclassical ones and can describe the evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. These asymptotics represent small perturbations on the background of an exact quasi-stationary solution. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as pattern formation.

Abstract:
The Killingbeck potential consisting of the harmonic oscillator-plus-Cornell potential, is of great interest in high energy physics. The solution of Dirac equation with the Killingbeck potential is studied in the presence of the pseudospin (p-spin) symmetry within the context of the quasi-exact solutions. Two special cases of the harmonic oscillator and Coulomb potential are also discussed.

A lot of methods, such as Jacobian elliptic function analysis, are used to look for the explicit exact solution of Duffing differential equation. The key of the analysis is to construct quotient trigonometric function, and then nonlinear algebraic equation set theory and method are used for the solution of some kinds of nonlinear Duffing differential equation. In this paper, the exact solution of Duffing equation is obtained by using constant variation method, making use of the formula to solve cubic equations and general solution of the homogeneous equation of Duffing equation with appropriate Constant m and function f(t) .

Abstract:
Existence and uniqueness of solutions for equations is preliminary and foundation to study the behavior and property of solutions. The extended Fisher- Kolmogorov(EFK) equation plays an important role in the study of pattern formation in bi-stable systems, general genetics, the spread of liquid in the domain wall and traveling wave of reaction-diffusion system. In this paper, two-dimensional Extended Fisher-Kolmogorov(EFK) equation can be renormalized into ordinary differential equations by using Galerkin truncated method, and the existence and uniqueness of solutions for initial value problem of ordinary differential equations is proved. According to energy estimates of trucated solution in corresponding functional space, the convergence of truncated solution in corresponding functional space is given and the existence of weak solutions for two-dimensional Extended Fisher-Kolmogorov(EFK) equation is proved.Lastly, uniqueness for the weak solutions of two dimensional Extended Fisher-Kolmogorov(EFK) equation is derived in the condition that f (u) satisfies the Lipschitz conditions with repect to u .

Abstract:
The vacuum field equation in Finsler spacetime is equivalent to vanishing of Ricci scalar. We present an exact solution of the Finslerian vacuum field equation. The solution is similar to the Schwarzschild metric. It reduces to Schwarzschild metric while the Finslerian parameter vanishes. We get solutions of geodesic equation in such a Schwarzschild-like spacetime, and show that the geodesic equation returns to the counterpart in Newton's gravity at weak field approximation. It is proved that the Finslerian covariant derivative of the geometrical part of the gravitational field equation is conserved. The interior solution is also given.

Abstract:
An analytical solution of the selfconsistent Vlasov equation is presented. The time evolution is entirely determined by the initial distribution function. The largest Lyapunov exponent is calculated analytically. For special parameters of the model potential positive Lyapunov exponent is possible. This model may serve as a check for numerical codes solving selfconsistent Vlasov equations. The here presented method is also applicable for any system with analytical solution of the Hamilton equation for the formfactor of the potential.