Abstract
Using the semi-inverse method, a variational formulation is established for the Boussinesq wave equation. Based on the obtained variational principle, solitary solutions in the sech-function and expfunction forms are obtained.
Introduction
The search for the exact travelling wave solutions to nonlinear disperse wave systems plays an important role in the study of nonlinear physical phenomena. In the past few years, many effective methods have been suggested for searching for soliton solutions for various nonlinear wave equations, among which the variational iteration method [1 -5] , the homotopy perturbation method [6 -10] , the parameter-expansion method [11 -13] , He's variational method [8, 14, 15] and the exp-function method [16 -20] have been shown to be effective, easy, and accurate for a large class of nonlinear problems. D'Acunto [21, 22] applied He's variational method to the determination of limit cycles. Ozis and Yildirim [23] , Zhang [24] , and the present author [25] illustrated that He's variational method was also valid for nonlinear wave equations.
Introduction
In this paper, we will consider the Boussinesq wave equation [26] 
Introduction
which was proposed by Boussinesq to describe surface water waves whose horizontal scale is much larger than the depth of the water. The equation also arises in other physical applications such as nonlinear lattice waves, iron sound waves in a plasma, vibrations in a nonlinear string, and in the percolation of water in porous subsurface strata [27] . the tanh-method. Javidi and Jalilian in [27] apply the variational iteration method to construct solitary wave solutions of (1). The aim of this paper is to couple He's semiinverse method [14] and the exp-function method [16] to search for solitary solutions of the discussed problem.
Variational Formulation
According to He's variational method [8] for solitary solutions, a variational formulation should be first established.
Variational Formulation
For convenience, we seek the travelling wave solution of the Boussinesq wave equation by assuming the solution in the following form:
Variational Formulation
where c is a constant to be determined. Substitution the solution (2) into (1) gives
Variational Formulation
where the prime expresses the derivative with respect to ξ . Integrating (3) twice, we have
Variational Formulation
where g 1 and g 2 are the constants of integration. We set g 1 = g 2 = 0 for simplicity, then (4) reduces to
Variational Formulation
The variational iteration method and the homotopy perturbation method can be applied to (5) effectively.
Variational Formulation
Here we follow He's variational method described in [8] , where the Korteweg-de Vries (KdV) equation was used to illustrate the effectiveness and convenience of the suggested method. According to [8] , a variational formulation should be first established using the semi-inverse method [26] :
Variational Formulation
3. Soliton Wave Solution
The Form of Sech-Function
where p and q are unknown constants to be further determined. Substituting (7) into (6), we have
The Form of Sech-Function
Making J stationary with respect to p and q results in
The Form of Sech-Function
or simplified
The Form of Sech-Function
Solving (11) and (12) simultaneously results in
The Form of Sech-Function
The solitary solution is, therefore, obtained as follows:
The Form of Sech-Function
for a > 0 and c 2 < 1. This is exactly the soliton solution obtained in [26] .
The Form of Simple Exp-Function
As mentioned above the exp-function method [16] is very simple and straightforward, which is based on the assumption that travelling wave solutions can be expressed in the following exp-function form:
The Form of Simple Exp-Function
, where c, d, p, and q are positive integers which are unknown to be further determined, a n and b m are unknown constants. In this paper we use a simplified expfunction method, i. e., we search for a solitary solution in the following simple exp-function form:
The Form of Simple Exp-Function
where k and m are unknown constants to be further determined.
The Form of Simple Exp-Function
Substituting (15) into (6) and using the integration relations
The Form of Simple Exp-Function
we have
The Form of Simple Exp-Function
Making J stationary with respect to k and m results in 
The Form of Simple Exp-Function
Solving (21) and (22) simultaneously results in
The Form of Simple Exp-Function
The solitary solution is, therefore, obtained as follows:
The Form of Simple Exp-Function
To our knowledge the above form solitary solution is first appeared in literature. Figures 1 and 2 show the plots of 
Conclusion
To summarize, we can conclude from the obtained results that the variational approach to the Boussinesq wave equation has been demonstrated successfully. This method is extremely simple in its principle and quite easy to use. We predict the method will become a powerful mathematical tool to find solitary solutions of different forms for various nonlinear equations.

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