Solitary wave solution for the generalized Kawahara equation

The travelling wave ansatz is used to find the solitary wave solution of the generalized Kawahara equation. The ansatz is obtained from the structure of the soliton solution of the Kawahara equation and the modified Kawahara equation. The first two integrals of motion of the generalized Kawahara equation are also computed in this work.     

分数阶广义扰动热波方程的与泛函映射解

研究了一类分数阶广义非线性扰动热波方程.首先在典型分数阶热波方程情形下得到解,接着用泛函分析映射方法,求出了分数阶广义非线性扰动热波方程初始边值问题的任意次近似解析解.最后简述了它的物理意义.求得的近似解析解,弥补了单纯用数值方法得到的模拟解的不足.     

Solitary wave solution for the generalized KdV equation with time-dependent damping and dispersion

The solitary wave solution of the generalized KdV equation is obtained in this paper in presence of time-dependent damping and dispersion. The approach is from a solitary wave ansatze that leads to the exact solution. A particular example is also considered to complete the analysis.     

A solution to an open problem for wave equations with generalized Wentzell boundary conditions

In this paper we solve an open problem put forward by A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, concerning the mixed problem for wave equations with generalized Wentzell boundary conditions. As a consequence, we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized Wentzell boundary conditions. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, Tübingen, Germany (e-mail: tixi@fa.uni-tuebingen.de;jili@fa.uni-tuebingen.de)The first author acknowledges support from the Alexander-von-Humboldt Foundation and from CAS and NSFC. The second author acknowledges support from the Max-Planck Society and from CAS and EMC.     

修正的广义Vakhnenko方程的周期解

利用Hirota双线性方法以及Rienmann theta函数,构造了含两个任意常系数的修正的广义Vakhnenko方程的周期解.特别是在极限情况下,可以由方程的周期解得到其孤子解.     

Approximate traveling wave solution for shallow water wave equation

Reconstruction of Variational Iteration Method (RVIM) is used for computing the coupled Whitham-Broer-Kaup shallow water. Then RVIM solution is verified against exact one and is compared with powerful approximate solutions, the Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). The existent error of the methods is computed and convergence of the RVIM solution has been presented. Results obtained expose effectiveness and capability of this method to solve the nonlinear systems in mechanics, analytically.     

Periodic solution and wave front solution for delay equation

First, a general theorem on the existence of periodic solutions for equations with small discrete delay is obtained by employing a technique which is based on a result of the existence of inertial manifold for small discrete delay equation, meanwhile, this general theorem is applied to show the existence of periodic solution for a predator–prey system with small discrete delay. Second, this technique is also used to obtain the existence of travelling wave solution for a host–vector disease model with small discrete delay.     

Approximate method for the solution of the generalized Dirichlet problem

We extend the method for approximate solution of classical boundary-value problems for the Laplace equation suggested in [1–3] to the case of the Poisson equation with generalized functions on the right-hand side of the equation and in the boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.46, No. 10, pp. 1417–1420, October, 1994.     

Bifurcation of travelling wave solutions for the generalized ZK equations

By using the bifurcation theory of planar dynamical systems to the generalized ZK equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Instability of solitary wave solutions for the generalized BO-ZK equation

In this paper we study the generalized BO-ZK equation in two space dimensions

ut+upux+αHuxx+εuxyy=0.     

The Cauchy problem for the generalized hyperelastic rod wave equation

In this paper we consider the Cauchy problem for the generalized hyperelasticrod wave equation which includes the Camassa‐Holm equation and the hyperelastic rod wave equation. Firstly, by using the Kato's theory, we prove that the Cauchy problem for the generalized hyperelastic rod wave is locally well‐posed in Sobolev spaces with . Secondly, we give some conservation laws, some useful conclusions and the precise blow‐up scenario and show that the Cauchy problem for the generalized hyperelastic rod wave equation has smooth solutions which blows up in finite time. Thirdly, we give the blow‐up rate of the strong solutions to the Cauchy problem for the generalized hyperelastic rod wave equation. Finally, we give the lower bound of the maximal existence time of the solution and the lower semicontinuity of existence time of solutions to the generalized hyperelastic rod wave equation.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.     

Bifurcation of travelling wave solutions for the generalized KP-MEW equations

By using the theory of bifurcations of planar dynamical systems to the generalized KP-MEW equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.