Direct Approach to Construct the Periodic Wave Solutions for Two Nonlinear Evolution Equations

With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.     

Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.     

The Periodic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobi elliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.     

Binary Bell Polynomials,Bilinear Approach to Exact Periodic Wave Solutions of (2+1)-Dimensional Nonlinear Evolution Equations

In the present letter, we get the appropriate bilinear forms of(2+1)-dimensional KdV equation, extended (2+1)-dimensional shallow water wave equation and (2+1)-dimensional Sawada-Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.     

On Triply Periodic Wave Solutions for(2+1)-Dimensional Boussinesq Equation

By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1 /6)(u0 1) + 2[ln f(x,y,t)] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.     

Solitary Wave and Non-traveling Wave Solutions to Two Nonlinear Evolution Equations

By applying the extended homogeneous balance method, we find some new explicit solutions to two nonlinear evolution equations, which include n-resonance plane solitary wave and non-traveling wave solutions.     

Solitary Wave and Non-traveling Wave Solutions to Two Nonlinear Evolution Equations

By applying the extended homogeneous balance method, we find some new explicit solutions to two nonlinear evolution equations, which include n-resonance plane solitary wave and non-traveling wave solutions.     

The Peridic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobielliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions andthe other type of traveling wave solutions for the system are obtained.     

New Periodic Wave Solutions to Generalized Klein-Gordon and Benjamin Equations

By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.     

Periodic Solutions for a Class of Nonlinear Differential-Difference Equations

In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for three nonlinear differential-difference equations are obtained.     

Periodic Wave,Solitary Wave and Compacton Solutions of a Nonlinear Wave Equation with Degenerate Dispersion

In this letter, we investigate traveling wave solutions of a nonlinear wave equation with degenerate dispersion. The phase portraits of corresponding traveling wave system are given under different parametric conditions. Some periodic wave and smooth solitary wave solutions of the equation are obtained. Moreover, we find some new hyperbolic function compactons instead of well-known trigonometric function compactons by analyzing nilpotent points.     

A New Approach to Solve Nonlinear Wave Equations

From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that more new periodic solutions can be obtained by this new approach, and more shock wave solutions or solitary wave solutions can be got under their limit conditions.     

Periodic Solutions for Two Coupled Nonlinear-Partial Differential Equations

In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.     

Periodic Solutions for Two Coupled Nonlinear-Partial Differential Equations

In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and 0, the hyperbolic function solutions (including the solitary wave solutions) and trigonometric solutions are also given respectively.     

Applications of F-expansion to Periodic Wave Solutions for Variant Boussinesq Equations

We present an F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the variant Boussinesq equations. When the modulus m approaches 1 and O, the hyperbolic function solutions （including the solitary wave solutions） and trigonometric solutions are also given respectively.     

Quasi-Periodic Waves and Asymptotic Property for Boiti-Leon-Manna-Pempinelli Equation

In this paper, multi-periodic （quasi-periodic） wave solutions are constructed for the Boiti-Leon-Manna- Pempinelli （BLMP） equation by using Hirota bilinear method and Riemann theta function. At the same time, we analyze in details asymptotic properties of the multi-periodic wave solutions and give their asymptotic relations between the periodic wave solutions and the soliton solutions.     

Elliptic Equation and New Solutions to Nonlinear Wave Equations

The new solutions to elliptic equation are shown, and then the elliptic equation is taken as a transformationand is applied to solve nonlinear wave equations. It is shown that more kinds of solutions are derived, such as periodicsolutions of rational form, solitary wave solutions of rational form, and so on.     

Elliptic Equation and Its Direct Applications to Nonlinear Wave Equations

Elliptic equation is taken as an ansatz and applied to solve nonlinear wave equations directly. More kinds of solutions are directly obtained, such as rational solutions, solitary wave solutions, periodic wave solutions and so on. It is shown that this method is more powerful in giving more kinds of solutions, so it can be taken as a generalized method.