ABSTRACT

By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation terms in.

Keywords: Complete Discrimination System for Polynomial; Traveling Wave Solution; Generalized Strong Nonlinear Boussinesq Equation without Dissipation Terms

1. Introduction

There are many methods of obtaining the exact solutions for nonlinear evolution equations, such as the homogeneous balance method [1], the inverse scattering method [2], Hirotas bilinear transformation [3], the extended tanh-function method [4], the sech-function method [5] and so on. Liu introduced complete discrimination system for the polynomial method to obtain the classification of traveling wave solutions to some nonlinear evolution equations [6-8]. In [9], the generalized strong nonlinear Boussinesq equation without dissipation terms was given by

(1)

where, , are constants. When, Equation (1) becomes

(2)

Equation (2) is an important model equation in physics. It describes the wave propagation in the weakly nonlinear and dispersive media. When or, the Equation (2) becomes good Boussinesq equation [10,11] or bad Boussinesq equation [12,13]. The good Boussinesq equation and bad Boussinesq equation have been studied by many authors [10-17]. But the classification of single traveling wave solutions to these equations hasn't been studied. In the present paper, we consider the following generalized strong nonlinear Boussinesq equation without dissipation terms in:

(3)

where, are constants. By using Liu’s method, the classification of single traveling wave solutions to Equation (3) is obtained.

2. The Traveling Wave Solutions to the Equation (3)

Take wave transformation

(4)

Substituting Equation (4) into Equation (3) yields the following nonlinear ordinary difference equation:

(5)

Integrating Equation (5) once with respect to, and setting the integration constant to zero yields:

(6)

Integrating Equation (6) twice with respect to yields:

(7)

where and are arbitrary constants.

In order to find the traveling wave solutions to the Equation (3), let us solve Equation (7). In this article, there are two cases to discuss the exact solutions of Equation (7) according to the arbitrary constant.

Case 2.1, then Equation (7) becomes

(8)

Integrating Equation (8) once yields

(9)

where

(10)

If, we take; if, we take. The complete discrimination system for the third order polynomial is given as follows:

(11)

In order to obtain the solutions to the Equation (9), according to the complete discrimination system for the third order polynomial, there are four cases to be discussed.

Case 2.1.1., where are real constants, , If

, when and, from Equation (9), we give the solution of Equation (7) as follows:

(12)

when and, we have

(13)

when, we have

(14)

If, when and, from Equation (9), we give the solutions of Equation (7)

(15)

when, and, we have

(16)

when, we have

(17)

Case 2.1.2. , where is real constant. If, when, we have

(18)

If, when, we have

(19)

Case 2.1.3., where are different real constants. If

, when, we have

(20)

(21)

where. If, when, we have

(22)

(23)

where.

Case 2.1.4., , where are all real constants, andwe have

(24)

where

,.

Case 2.2 In order to solve Equation (7), when, we take the transformation as follows

(25)

Combining the expression (7) with Equation (25) yields

(26)

where

, And

When, we take the following transformation:

(27)

Combining the expression (7) with Equation (27) yields

(28)

where

, and

The complete discrimination system for the fourth order polynomial as follows:

(29)

In order to obtain the solutions to Equation (26) and Equation (28), according to the complete discrimination system for the fourth order polynomial, there are nine cases to be discussed.

Case 2.2.1, , and, then. where. For, the solution of Equation (7) is

(30)

Case 2.2.2. and then For, the solution of Equation (7) is

(31)

Case 2.2.3. and, , where Forwhen or, the solution of Equation (7) is

(32)

when,the solution of Equation (7) is

(33)

Case 2.2.4. and, then when,

or, the solution of Equation (7) is

(34)

when, or, the solution of Equation (7) is

(35)

Case 2.2.5 and, then If, when

and or when and, we have

(36)

when and, or when, and, we have

(37)

when, we have

(38)

If, when and or when and, we have

(39)

when and, or when, and, we have

(40)

when, we have

(41)

where, and.

Case 2.2.6. and,

where If when or, the solution of Equation (7) is

(42)

when the solution of Equation (7) is

(43)

If when the solution of Equation (7) is

(44)

when the solution of Equation (7) is

(45)

where.

Case 2.2.7. and,. where and.

The solution of Equation (7) (when, we take the positive sign; when we take the negative) is

(46)

where, , ,

, ,.

Case 2.2.8. and, then, where. The solution of Equation (7) is

(47)

where, , ,

,.

Case 2.2.9. and, then, where and are real numbers. If, we have

(48)

where.

4. Conclusion

By the complete discrimination system for polynomial method, we have obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq without dissipation terms in .These solutions include trigonometric periodic solutions, rational function solution, hyperbolic funtion solutions, Jacobi elliptic function solutions and so on. This method is simple and efficient.

Acknowledgements

The project is supported by Scientific Research Fund of Education Department of Heilongjiang Province of China under Grant No. 12521049.

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NOTES

^*Corresponding author.