1. Introduction

The study of exact solutions of nonlinear partial differential equations plays an important role in soliton theory and explicit formulas of nonlinear partial differential equations play an essential role in the nonlinear science. Also, the explicit formulas may provide physical information and help us to understand the mechanism of related physical models. Recently, many kinds of powerful methods have been proposed to find exact solutions of nonlinear partial differential equations, e.g., the tanh-method[1], the homogeneous balance method[2], homotopy analysis method[3-8], the expansion method [9], three-wave method[10-13], extended homoclinic test approach[14-16], the expansion method[17] and the exp-function method[18-23].

By these analytic methods, one only can obtain traveling wave solutions for NLPDEs. However, it is known that there are multiple wave solutions to some NLPDEs, for example, multi-soliton solutions to KdV and Toda lattice equations[24] or multiple periodic wave solutions to the Hirota bilinear equations[25,26]. Recently, Ma and Fan[27] explored a key feature of the linear superposition principle that linear equations possess, for Hirota bilinear equations, while aiming to construct a specific sub-class of -soliton solutions formed by linear combinations of exponential traveling waves. They proved that, a linear superposition

principle can apply to exponential traveling waves of Hirota bilinear equations. Applications made to show that the presented linear superposition principle is helpful in generating-wave solutions to soliton equations, particularly those in higher dimensions. Also, Ma et al.[28] constructed the multiple exp-function method to obtain multiple wave solutions, including one-soliton, two-soliton and three-soliton type solutions to (3+1)-dimensional YTSF equation. One may find another works to find exact solutions of soliton equations in[29-31].

In this paper, we apply the multiple exp-function method to obtain some exact multiple wave solutions for the (2 + 1)- and (3 + 1)-dimensional breaking soliton equations.

The (2+1)-dimensional breaking soliton equation is

(1)

this equation describes the (2+1)-dimensional interaction of the Riemann wave propagated along the -axis with a long wave propagated along the -axis[32]. Wazwaz[33] introduced an extension to equation (1) by adding the last three terms with replaced by . His work, enables us to establish the following (3+1)-dimensional breaking soliton equation

(2)

where

This paper is organized as follows: in the following section we have a brief review on multiple exp-function method. In Sections 3 and 4 we apply the exp-function method on the (2 + 1)-dimensional and the (3 + 1)-dimensional breaking soliton equations, respectively. In those sections we obtain one-wave, two-wave and three-wave solutions for our equations. The paper is concluded in Section 4.

2. Methodology

Ma et al.[28] illustrated the multiple exp-function method in three steps. In summary, their steps are: Defining solvable differential equations, transforming nonlinear PDEs and solving algebraic systems. The key point of their approach is to seek rational solutions in a set of new variables defining individual waves. The application of multiple exp-function method yields specific one-wave, two-wave and three-wave solutions to NLPDEs. To explain these fundamental steps in multiple exp-function method, consider a PDE in (1 + 1) dimensions as

(3)

In the first step, we must introduce a sequence of new variables , by solvable partial differential equations, for example, the linear PDEs,

(4)

where are the angular wave numbers and , are the wave frequencies. This step is a starting point to construct exact solutions of nonlinear equations. Solving these linear equations leads to the exponential function solutions,

(5)

where are arbitrary constants. Any function describing a single wave and a multiple wave solution will be a combination of all of those single waves. In the second step, we proceed by considering rational solutions in the new variables

(6)

where and are constants to be determined from (3). Then we use the differential relations in (4) to express all partial derivatives of with and in terms of Substituting these obtained expressions for partial derivatives into the equation (3) generates a rational function equation in the new variables :

(7)

Equation (7) is called the transformed equation of the original equation (3). This step makes it possible to compute solutions to differential equations directly by computer algebra systems in the third step. Finally, in the third step we set the numerator of the resulting rational function to zero. This yields a system of algebraic equations on all variables By solving this system, with the aid of a mathematical software such as Maple, we determine polynomials and and the wave exponents . After this, the multiple wave solution is computed and given by

(8)

for more details on these steps; cf.[28].

In the following sections we apply multiple exp-function method on the (2 + 1)- and the (3 + 1)-dimensional breaking soliton equations to obtain some multi-wave solutions. It must be noted that all obtained solutions are new ones and for some special values of parameters in these new solutions we can obtain the solutions which have obtained by another methods.

3. One-Wave, Two-Wave and Three- Wave Solutions to the (2 + 1) -Dimensional Breaking Soliton Equation

In this part, we apply the multiple exp-function method to construct the exact multiple wave solutions of the (2 + 1) -dimensional breaking soliton equation

(9)

We generate one-wave, two-wave and three-wave solutions of two polynomial functions and as follows:

Case i: One-wave solutions

To obtain one-wave solutions, we require the linear conditions

(10)

where and are constants which will be determined. Then we try a pair of polynomials of degree one,

(11)

where and are constants to be determined. By the multiple exp-function method and using the differential relations in (10), we obtain the following solution to the resulting algebraic system with the aid of Maple,

(12)

and are arbitrary. Thus we can have an exponential function solution to (10),

(13)

which gets the following one-wave solutions

(14)

where and are arbitrary constants. It must be noted that, if we set and , the solution (19) of[33] will obtain.

Case ii: Two-wave solutions

In this case, we require the linear conditions,

(15)

where , , are constants and thus the solutions and are defined by

(16)

where and are arbitrary constants.

We suppose that

(17)

where is a constant to be determined. By the multiple exp-function method and using the differential relations in (15), we obtain the following solution to the resulting algebraic system with the aid of Maple,

(18)

and

(19)

Then, we obtain the following two-wave solutions

(20)

Figure 1. The two-wave solution

For , solution (20) is as same as solution (24) of[33].

We obtained a wide class of traveling wave solutions to Eq. (9) by setting special values to the arbitrary parameters, we can construct formal two-wave solutions. For instance, taking and in (20) reveals solitary waves to Eq. (9). Figure 1 shows the plot of solution (20) for these special values of its parameters.

Case iii: Three-wave solutions

Similarly, to obtain the three-wave solutions for equation (9), we require the linear conditions,

(21)

(22)

whereand are arbitrary constants.We suppose that

(23)

where and are constants to be determined. By the multiple exp-function method and using the differential relations in (21), we obtain the following solution to the resulting algebraic system with Maple,

(24)

and

(25)

Then, we obtain the following three-wave solutions

(26)

whereand are defined by (23) and are defined by (22). For in (26), we meet solution (26) of [32].

Figure 2 shows the plot of solution (26) for some special values of its parameters.

4. One-Wave, Two-Wave and Three- Wave Solutions to the (3 + 1) -Dimensional Breaking Soliton Equation

In this section, we will apply the multiple exp-function method to construct the exact multiple wave solutions for the (3 + 1)-dimensional breaking soliton equation

(27)

we discuss three cases of two polynomial functions and to generate one-wave, two-wave and three-wave solutions as follows:

Case i: One-wave solutions

Again, we require the linear conditions,

(28)

where are constants. We then try a pair of polynomials of degree one,

(29)

where and are constants to be determined. By the multiple exp-function method and using the differential relations in (28), we obtain the following solution to the resulting algebraic system with the help of Maple,

(30)

where and are arbitrary constants. Since we can have an exponential function solution to (28),

(31)

By using (31), we obtain the following one-wave solutions for equation (27)

Figure 2. The three-wave solution.

(32)

where and are arbitrary constants. For and our solution (32) is the same solution (49) of [33].

Case ii: Two-wave solutions

In this case, we require the linear conditions,

(33)

where , are constants and thus the solutions and can be defined by

(34)

where and are arbitrary constants.

We suppose that

(35)

Figure 3. The first two-wave solution.

where is a constant to be determined. By the multiple exp-function method and using the differential relations in (33), we obtain two solutions to the resulting algebraic system with Maple,

(36)

and

(37)

when , and

(38)

and

(39)

when we obtain the following two-wave solutions

(40)

Two specific solutions of these two-wave solutions are plotted in Figures 3 and 4.

Figure 4. The second two-wave solution.

Case iii: Three-wave solutions

To obtain the three-wave solutions for equation (27), we require the linear conditions,

(41)

where , , are constants and thus the solutions and can be defined by

(42)

where and are arbitrary constants.

We suppose that

(43)

where and are constants to be determined. By the multiple exp-function method and using the differential relations in (41), we obtain the following solution to the resulting algebraic system with the help of Maple,

(44)

and

(45)

Figure 5. The first three-wave solution.

Figure 6. The second three-wave.

when , and

(46)

and

(47)

when .Then, we obtain the following three-wave solutions for equation (27)

(48)

Two specific solutions of these three-wave solutions are plotted in Figures 5 and 6.

5. Conclusions

In this paper, we have applied the multiple exp-function method to obtain one-wave, two-wave and three-wave solutions of the (2 + 1)- and the (3 + 1)-dimensional breaking soliton equations. These solutions are new solutions for these equations and for some special values of parameters in these new solutions we can obtain the solutions which have obtained by another methods. The multiple exp-function method is oriented towards the ease of use and capability of computer algebra systems and provides a straightforward and systematic solution procedure that generalizes Hirota's perturbation method. The method can be applied on other nonlinear partial differential equations which have multi-soliton solutions, because the multiple exp-function algorithm is powerful in generating exact solutions of nonlinear equations.