Variant Boussinesq Equations Research Articles

(G′/G)-Expansion Method Equivalent to Extended Tanh Function Method

In a recent article [Physics Letters A 372 (2008) 417], Wang et al. proposed a method, which is called the (G′/G)-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota–Satsuma equations are obtained by using this method. They think the (G′/G)-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the (G′/G)-expansion method is equivalent to the extended tanh function method.

The ()-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics

The ( G ′ G )-expansion method is firstly proposed, where G = G ( ξ ) satisfies a second order linear ordinary differential equation (LODE for short), by which the travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations and the Hirota–Satsuma equations are obtained. When the parameters are taken as special values the solitary waves are also derived from the travelling waves. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The proposed method is direct, concise, elementary and effective, and can be used for many other nonlinear evolution equations.

Extended Fan's Algebraic Method and Its Applicationto KdV and Variant Boussinesq Equations

An extended Fan's algebraic method is used forconstructing exact traveling wave solution of nonlinear partial differentialequations. The key idea of this method is to introduce an auxiliary ordinarydifferential equation which is regarded as an extended elliptic equation andwhose degree r is expanded to the case of r>4. The efficiency of themethod is demonstrated by the KdV equation and the variant Boussinesqequations. The results indicate that the method not only offers allsolutions obtained by using Fu's and Fan's methods, but also some new solutions.

EXP-function method and its application to nonlinear equations

Exp-function method is used to find a unified solution of a nonlinear wave equation. Variant Boussinesq equations are selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free parameters is obtained.

Exact travelling wave solutions for two nonlinear evolution equations using the improved [formula omitted]-expansion method

In this letter, an improved F -expansion method with a computerized symbolic computation is used to derive the new exact travelling wave solutions expressed by Jacobi elliptical functions for nonlinear evolution equations. The validity and reliability of the method is tested by its applications for two nonlinear evolution equations, namely, coupled ZK equations and variant Boussinesq equations. The method is straightforward, concise and is a promising and powerful method for other nonlinear evolution equations in mathematical physics. As a result, many different classes of solutions are obtained in terms of Jacobi elliptical functions, Weierstrass elliptical function, triangle functions, hyperbolic functions and other types of functions. In limiting cases, solitary wave solutions of the equations are also obtained.

Bifurcations of travelling wave solutions in variant Boussinesq equations

The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.

An Improved Modified Extended tanh-Function Method

In this paper we further improve the modified extended tanh-function method to obtain new exact solutions for nonlinear partial differential equations. Numerical applications of the proposed method are verified by solving the improved Boussinesq equation and the system of variant Boussinesq equations. The new exact solutions for these equations include Jacobi elliptic doubly periodic type,Weierstrass elliptic doubly periodic type, triangular type and solitary wave solutions

An Explicit Bäcklund Transformation of BurgersEquation with Applications**The project supported by ChinaPostdoctoral Science Foundation under Grant No. 2004036086,K.C. Wong Education Foundation, Hong Kong, andpartially supported by the National Key Basic Research Projectof China under Grant No. 2004CB318000

In this paper, an explicit Bäcklund transformation (BT) of theBurgers equation is obtained by using the further extended tanhmethod [Phys. Lett. A 307 (2003) 269; Chaos, Solitons & Fractals17 (2003) 669]. Based on the BT and some newly obtained seedsolutions, infinite sequences of exact solutions for the Burgersequation are generated. Further more, this BT of the Burgersequation is applied to solve the variant Boussinesq equations andthe approximate equations of long water wave.

Applications of F-expansion to Periodic Wave Solutions for VariantBoussinesq Equations**The project supported in part by the Natural ScienceFoundation of Henan Province of China under Grant No. 0111050200,the Natural Science Foundation of Education Department of HenanProvince of China under Grant No. 2003110003, and the ScienceFoundation of Henan University of Science and Technologyunder Grant No. 2004ZD002

We present an F-expansion method forfinding periodic wave solutions of nonlinear evolution equationsin mathematical physics, which can be thought of as aconcentration of extended Jacobi elliptic function expansionmethod proposed recently. By using the F-expansion, withoutcalculating Jacobi elliptic functions, we obtain simultaneouslymany periodic wave solutions expressed by various Jacobi ellipticfunctions for the variant Boussinesq equations. When the modulusm approaches 1 and 0, the hyperbolic function solutions(including the solitary wave solutions) and trigonometricsolutions are also given respectively.

The extended Fan's sub-equation method and its application to KdV–MKdV, BKK and variant Boussinesq equations

An extended Fan's sub-equation method is used for constructing exact travelling wave solutions of nonlinear partial differential equations (NLPDEs). The key idea of this method is to take full advantage of the general elliptic equation involving five parameters which has more new solutions and whose degeneracies can lead to special sub-equations involving three parameters. More new solutions are obtained for KdV–MKdV, Broer–Kaup–Kupershmidt (BKK) and variant Boussinesq equations. Then we present a technique which not only gives us a clear relation among this general elliptic equation and other sub-equations involving three parameters (Riccati equation, first kind elliptic equation, auxiliary ordinary equation, generalized Riccati equation and so on), but also provides an approach to construct new exact solutions to NLPDEs.

Bifurcations of traveling wave solutions for two coupled variant Boussinesq equations in shallow water waves

Abstract The bifurcations of traveling wave solutions for two coupled variant Boussinesq equations introduced as a model for water waves are studied in this paper. Transition boundaries have been presented to divide the parameter space into different regions associated with qualitatively different types of solutions. The conditions for the existence of solitary wave solutions and uncountably infinite, smooth, non-smooth and periodic wave solutions are obtained. The explicit exact traveling wave solutions are presented by using an algebraic method.

Jacobi elliptic function solutions for two variant Boussinesq equations

Abstract A general Jacobi elliptic function expansion method is proposed to construct abundant Jacobi elliptic function (doubly periodic) solutions for two variant Boussinesq equations. These Jacobi elliptic function solutions degenerate to the soliton wave solutions and trigonometric function solutions at a certain limit condition.

The new constructive algorithm and symbolic computation applied to exact solutions of nonlinear wave equations

In this Letter, a new method (called generalized sine-Gordon equation expansion method) is proposed by further studying the famous sine-Gordon equation and using a generalized transformation to seek more types of solutions of nonlinear wave equations. With the aid of symbolic computation, we choose ( 2 + 1 ) -dimensional Burgers equation and the variant Boussinesq equations to illustrate the validity and advantages of the algorithm. As a consequence, more types of new solitary wave solutions, singular solitary wave solutions and doubly periodic solutions are obtained. The algorithm can be also extended to many other nonlinear wave equations.

Evolution property of soliton solutions for the Whitham-Broer-Kaup equation and variant Boussinesq equation

Using the standard Painleve analysis approach, the (1+1)-dimensional Whitham-Broer-Kaup (WBK) and variant Boussinesq equations are solved. Some significant and exact solutions are given. We investigate the behaviour of the interactions between the multi-soliton-kink-type solution for the WBK equation and the multi-solitonic solutions and find the interactions are not elastic. The fission of solutions for the WBK equation and the fusions of those for the variant Boussinesq equation may occur after their interactions.

Nonlinear Partial Differential Equations Solved by Projective Riccati Equations Ansatz

Based on the general projective Riccati equations method and symbolic computation, some new exact travelling wave solutions are obtained for a nonlinear reaction-diffusion equation, the highorder modified Boussinesq equation and the variant Boussinesq equation. The obtained solutions contain solitary waves, singular solitary waves, periodic and rational solutions. From our results, we can not only recover the known solitary wave solutions of these equations found by existing various tanh methods and other sophisticated methods, but also obtain some new and more general travelling wave solutions.

Generalized transformations and abundant new families of exact solutions for (2 + 1)-dimensional dispersive long wave equations

A new generalized transformation based upon the well-known Riccati equation is presented and applied to find exact solutions of the (2 + 1)-dimensional dispersive long wave equation. Many explicit exact solutions are obtained that include new solitary and periodic wave solutions, and combined formal solitary and periodic wave solutions. The variant Boussinesq equation, as a special case of the (2 + 1)-dimensional dispersive long wave equation, is also solved. These exact solutions are obtained computationally using the Wu elimination method with the aid of Mathematica to solve the large system of algebraic equations produced by the solution ansatz.

A REDUCTION mKdV METHOD WITH SYMBOLIC COMPUTATION TO CONSTRUCT NEW DOUBLY-PERIODIC SOLUTIONS FOR NONLINEAR WAVE EQUATIONS

Firstly twenty-four types of doubly-periodic solutions of the reduction mKdV equation are given. Secondly based on the reduction mKdV equation and its solutions, a systemic transformation method (called the reduction mKdV method) is developed to construct new doubly-periodic solutions of nonlinear equations. Thirdly with the aid of symbolic computation, we choose the KdV equation, the coupled variant Boussinesq equation and the cubic nonlinear Schrödinger equation to illustrate our method. As a result many types of solutions are obtained. These show that this method is simple and powerful to obtain more exact solutions including doubly-periodic solutions, soliton solutions and singly-periodic solutions to a wide class of nonlinear wave equations. Finally we further extended the method to a general form.

Mapping deformation method and its application to nonlinear equations

An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinear partial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraic mapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein-Gordon equation. This is applied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained, including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.

A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves

A new algebraic method is devised to uniformly construct a series of new travelling wave solutions for two variant Boussinesq equations. The solutions obtained in this paper include soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters.

New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water

Based upon the well-known Riccati equation, a new generalized transformation is presented and applied to solve Whitham–Broer–Kaup (WBK) equation in shallow water. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions and periodic wave solutions, are obtained. And variant Boussinesq equation and the system of approximate equation for long water waves, as the special cases of WBK equation, can also obtain the corresponding solitary wave solutions and periodic wave solutions. In addition, with the aid of Mathematica and Wu elimination method to solve a large system of algebraic equations, the course of solving equations can be carried out in computer.