Assessment of the further improved (G'/G)-expansion method and the extended tanh-method in probing exact solutions of nonlinear PDEs

PACS numbers 02.30.Jr, 05.45.Yv, 02.30.Ik

In this article, we bring in an alternative approach, called a further improved (G'/G)-expansion method to find the exact traveling wave solutions of the breaking soliton equation, where G = G(ξ) satisfies the auxiliary ODE [G′(ξ)] 2 = p G 2 (ξ) + q G 4 (ξ) + r G 6 (ξ); p, q and r are constants. Recently El-Wakil et al. (2010) and Parkes (2010) have shown that the extended tanh-function method proposed by Fan (2000) and the basic (G'/G)expansion method proposed by Wang et al. (2008) are entirely equivalent in as much as they deliver exactly the same set of solutions to a given nonlinear evolution equation. This observation has also been pointed out recently by Kudryashov (2009). In this article, we assert even though the basic (G'/G)-expansion method is equivalent to the extended tanh-function method, the further improved (G'/G)-expansion method presented in this letter is not equivalent to the extended tanhfunction method. The method projected in this article is varied to some extent from the extended (G'/G)expansion method.
The objective of this article is to show that the further improved (G'/G)-expansion method and the celebrated extended tanh-function method are not identical. Further novel solutions are achieved via the offered further improved (G'/G)-expansion method. It has not been used by somebody previously. This approach will play an imperative role in constructing many exact traveling wave solutions for the nonlinear PDEs via the (2 + 1)-dimensional breaking soliton equation.

The further improved (G'/G)-expansion method
Suppose we have the following nonlinear partial differential equation F u; u t ; u x ; u y ; u tt ; u x t ; u y t ; ⋯ À Á ¼ 0; where u = u(x, y, t) is an unknown function, F is a polynomial in u = u(x, y, t) and its partial derivatives in which the highest order derivatives and the nonlinear terms are involved. In the following we give the main steps of the further improved (G'/G)-expansion method.
Step 1: The traveling wave variable, where V is the speed of the traveling wave, permits us to convert the Eq. (1) into an ODE in the form, Step 2: Assume the solution of the Eq. (3) can be expressed by means of a polynomial in (G'/G) as follows: where α i (i = 1, 2, 3, ⋯) are constants provided α n ≠ 0 and G = G(ξ ) satisfies the following nonlinear auxiliary equation, Where p, q and r are random constants to be determined later (Table 1).
Step 3: In Eq. (4), n is a positive integer to be determined; typically this involves balancing the highest order nonlinear term(s) with the linear term(s) of the highest order come out in Eq. (3).
Step 4: Substituting Eq. (4), into Eq. (3) and utilizing Eq. (5), we obtain polynomials in G i (ξ ) and G'(ξ ) G i (ξ ) (i = 0, ± 1, ± 2, ± 3, ⋯). Vanishing each coefficient of the resulted polynomials to zero, yields a set of algebraic equations for α n , p, q, r, V and constant(s) of integration, if applicable. If the original evolution equation contains some arbitrary constant coefficients, these will, of course, also appear in the system of algebraic equations. Suppose with the aid of symbolic computation software such as Maple, the unknown constants α n , p, q, r and V can be found by solving these set of algebraic equations and substituting these values into Eq. (4), new and more general exact traveling wave solutions of the nonlinear partial differential Equation (1) can be found.

Application
In this section, we bring to bear the further improved (G'/G)-expansion method to the (2 + 1)-dimensional breaking soliton equation which is dreadfully important nonlinear evolution equations in mathematical physics and have been paid attention by a lot of researchers and the extended tanh-function method to compare the solutions obtained by the two methods.
On solving the (2 + 1)-dimensional breaking soliton equation by the projected method We start with the (2 + 1)-dimensional breaking soliton equation (Darvishi & Najafi 2012;Bekir 2010;Ma et al. 2009;Inan 2010) in the form, This equation was first introduced by Calogero and Degasperis in 1977. The breaking soliton equation describe the (2 + 1)-dimensional interaction of the Riemann wave propagation along the y-axis with a long wave propagation along x-axis (Ma et al. 2009). In the recent years, a considerable amount of research works on the breaking soliton equation have been accomplished. For example, its solitary wave solutions, periodic and multiple soliton solutions are found in (Inan 2010). Let us now solve the Eq. (6) by the proposed further improved (G'/G)-expansion method. To this end, we perceive that the traveling wave variable (2) permits us in converting Eq. (6) into an ODE and upon integration yields: with zero constant of integration. Considering the homogeneous balance between the highest order derivative and the nonlinear term come out in Eq. (7), we deduce that D(u′) 2 = D(u‴), where D(u′) 2 stands for degree of (u′) 2 and so on. This yield n = 1. Therefore, the solution (4) turns out to be Substituting (8) together with Eq. (5) into (7), we obtain the following polynomial equation in G: Setting each coefficient of the polynomial Eq. (9) to zero, we achieve a system of algebraic equations which can be solved by using the symbolic computation software such as Maple and obtain the following two sets of solutions: The set 1.
where α 0 , p and r are arbitrary constants. The set 2.
where α 0 , p and r are arbitrary constants.
Now for the set 1, we have the following solution: where ξ ¼ x þ y þ 4 p t: According to the step 2 of section 2, we have the subsequent families of exact solutions: Family 1. If p > 0, the solution of Eq. (5) has the form, Table 1 The general solutions of Eq. (5) are as follows (Yomba 2008;Zhang & Xia 2007) No In these cases we have the ratio, and respectively. Since q ¼ 2 ffiffiffiffiffi ffi p r p , subsequently, we obtain the following traveling wave solutions, Family 2. If p > 0, r > 0, the solution of Eq. (5) has the form, Then we have the ratio, Since q ¼ 2 ffiffiffiffiffi ffi p r p , subsequently, we obtain the following traveling wave solutions: Family 3. If p < 0, r > 0, the solution of Eq. (5) has the form, Then we have the ratio Since q ¼ 2 ffiffiffiffiffi ffi p r p , subsequently, we obtain the following traveling wave solutions: where ξ ¼ x þ y þ 4 p t: Family 4. If p > 0, Δ = 0, the solution of Eq. (5) has the form, Then we have the ratio Subsequently, we obtain the following traveling wave solutions: where ξ ¼ x þ y þ 4 p t Family 5. If p > 0, the solution of Eq. (5) has the form Then we have the ratio Since q ¼ 2 ffiffiffiffiffi ffi p r p , subsequently, we obtain the following traveling wave solutions: where For the set 2, we have the following solution: where ξ ¼ x þ y þ 16p t: According to the step 2 of section 2, we obtain the subsequent families of exact solutions: Cohort 1. If p > 0, Δ > 0, the solution of Eq. (5) has the form, Since q = 0, then r < 0. In this case we have the ratio, Therefore, we obtainthe following traveling wave solution, Cohort 2. If p > 0, Δ < 0, the solution of Eq. (5) has the form Since q = 0, then r > 0. In this case we have the ratio, Therefore, we obtain the following traveling wave solutions: Cohort 3. If p < 0, Δ > 0, the solutions of Eq. (5) has the form, although the extended tanh-function expansion method and the basic (G'/G)-expansion method are equivalent. We see that by means of the extended tanh-function we attain merely three solutions of the breaking soliton equation, alternatively through the further improved (G'/G)-expansion method we obtain twelve solutions of which three solutions are analogous, two are reducible and the rest of the seven solutions cannot be found by the extended tanh-function method. The calculations of the projected method are also easier than the extended tanh-function method as well as the basic (G'/G)-expansion method.

Conclusions
A further improved (G'/G)-expansion method is suggested and applied to the (2 + 1)-dimensional breaking soliton equation. The results obtained by the suggested method have been compared with those obtained by the celebrated extended tanh-function method. From this study, we observe that the further improved (G'/G)expansion method and the extended tanh-function method are not equivalent, although El-Wakil (El-Wakil et al. 2010) and Parkes (Parkes 2010) have shown that the basic (G'/G)expansion method and the extended tanh-function method are equivalent. We see that all the results obtained by the extended tanh-function are found by the suggested method and in addition some novel solutions are attained. It is evident that obtained solutions are more general and many known solutions are only special case of them. The analysis shows that the proposed method is quite resourceful and practically well suited to be used in finding exact solutions of NLEEs. We expect the suggested method might be applicable to other kinds of NLEEs in mathematical physics and this is our next job.