Traveling wave solutions of nonlinear evolution equations via the enhanced (G′/G)-expansion method

In this article, an enhanced (G′/G)-expansion method is suggested to find the traveling wave solutions for the modified Korteweg de-Vries (mKDV) equation. Abundant traveling wave solutions are derived, which are expressed by the hyperbolic and trigonometric functions involving several parameters. The efficiency of this method for finding these exact solutions has been demonstrated. It is shown that the proposed method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics.     

Exact traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation

Bifurcation method of dynamical systems is employed to investigate traveling wave solutions in the (2 + 1)-dimensional Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation. Under some parameter conditions, exact solitary wave solutions and kink wave solutions are obtained.     

Solitary wave solutions and kink wave solutions for a generalized KDV-mKDV equation

Bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions of the generalized KDV-mKDV equation. Under some parameter conditions, their explicit expressions are obtained.     

Exact Solitary Wave And Periodic Wave Solutions Of The Kaup-Kuperschmidt Equation

In this paper we investigate the exact traveling wave solutions of the fifth-order Kaup-Kuperschmidt equation. The bifurcation and exact solutions of a general first-order nonlinear equation are investigated firstly. With the help of Maple and by using the bifurcation and exact solutions of two derived subequations, we obtain two families of solitary wave solutions and two families of periodic wave solutions of the KK equation. The relationship of the two subequations and the two known rst integrals are analyzed.     

RLW-Burgers方程的一类解析解

本文给出了 RLW-Burgers方程及 Kd V-Burgers方程的一类解析解 ,且可得到 RLW-Burgers方程的振荡激波解 .这些解可以表示为 Burgers方程和 Kd V方程解的线性组合 ,文末还对文 [8]作了讨论 .     

New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics

In this article, new extension of the generalized and improved (G′/G)-expansion method is proposed for constructing more general and a rich class of new exact traveling wave solutions of nonlinear evolution equations. To demonstrate the novelty and motivation of the proposed method, we implement it to the Korteweg-de Vries (KdV) equation. The new method is oriented toward the ease of utilize and capability of computer algebraic system and provides a more systematic, convenient handling of the solution process of nonlinear equations. Further, obtained solutions disclose a wider range of applicability for handling a large variety of nonlinear partial differential equations.     

On the ZK equation with a directional dissipation

This paper studies the generalized Zakharov-Kuznetsov-Burgers equation. The initial value problem associated to this equation will be investigated in the nonhomogeneous Sobolev spaces and some suitable weighted spaces, under appropriate conditions. Moreover, an ill-posedness result (in some sense) will be proved in the anisotropic Sobolev spaces. Furthermore some exact traveling wave solutions of this equation will be obtained.     

On the traveling wave solutions for a parabolic–hyperbolic system in linear thermoelasticity

In this paper, the parabolic–hyperbolic system of linear thermoelasticity with variable coefficients is transformed into a system of two coupled equations. We discuss first the conditions which govern this separation in the case of a system of two coupled equations for which a general result on the separability is formulated. It is then shown that the explicit traveling wave solutions are obtained in the exact form.     

The homogeneous balance method and its application to the Benjamin-Bona-Mahoney (BBM) equation

We make use of the homogeneous balance method and symbolic computation to construct new exact traveling wave solutions for the Benjamin-Bona-Mahoney (BBM) equation. Many new exact traveling wave solutions are successfully obtained, which contain rational and periodic-like solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Solitary wave solutions and kink wave solutions for a generalized Zakharov-Kuznetsov equation

Bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions in the generalized Zakharov-Kuznetsov equation. Under some parameter conditions, their explicit expressions are obtained.     

A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order

In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations.     

Exact and explicit solutions to nonlinear evolution equations using the division theorem

In this paper, we show the applicability of the first integral method, which is based on the ring theory of commutative algebra, to the regularized long-wave Burgers equation and the Gilson-Pickering equation under a parameter condition. Our method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are derived in a concise manner.     

Solitary wave and chaotic behavior of traveling wave solutions for the coupled KdV equations

Using the method of dynamical systems to study the coupled KdV system, some exact explicit parametric representations of the solitary wave and periodic wave solutions are obtained in the given parameter regions. Chaotic behavior of traveling wave solutions is determined.     

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.     

Traveling Wave Fronts of a Degenerate Parabolic Equation with Non-divergence Form

We study the traveling wave solutions of a nonlinear degenerate parabolic equation with non-divergence form. Under some conditions on the source, we establish the existence, and then discuss the regularity of such solutions.     

Algebraic traveling waves for the generalized Newell–Whitehead–Segel equation

In this paper we provide the only possible algebraic traveling wave solutions for the celebrated general Newell–Whitehead–Segel equation.     

Traveling wave solutions for the mKdV equation and the Gardner equations by new approach of the generalized (G′/G)-expansion method

In this paper, we demonstrate the effectiveness of the new generalized (G′/G)-expansion method by seeking more exact solutions via the mKdV equation and the Gardner equations. The method is direct, concise and simple to implement compared to other existing methods. The traveling wave solutions obtained by this method are expressed in terms of hyperbolic, trigonometric and rational functions. The method shows a wide application for handling nonlinear wave equations. Moreover, the method reduces the large amount of calculations.