1. Introduction
The general form of the Drinfeld-Sokolov-Satsuma-Hirota system that is going to be studied in this paper is given by
(1)wt-6wwx+wxxx-6vx=0,vt-2vxxx+6wvx=0,
which was developed in [1] as one example of nonlinear equations possessing Lax pairs of a special form [2–4]. System (1) was independently presented by Drinfeld and Sokolov [1] and Satsuma and Hirota [5]. However, this system was found as a special case of the four-reduction of the KP hierarchy [2, 5]. Wazwaz [6] studied this system by using Hirota’s bilinear, the tanh-coth, the tan-cot, and the Exp-function methods. In [2], the truncated singular expansions method was used to construct an explicit Bäcklund transformation method to derive special solutions of this equation. Also, in [3], the sine-cosine method and the tanh method were used to obtain exact travelling wave solutions. Recently, the investigation of exact travelling wave solutions to nonlinear partial differential equations plays an important role in the study of nonlinear modelling physical phenomena. The study of the travelling wave solutions plays an important role in nonlinear sciences. Up to now, there exist many powerful methods to construct exact solutions of nonlinear differential equations. A variety of powerful methods have been presented, such as Hirota’s bilinear method [7–10], the inverse scattering transform [11], sine-cosine method [12], homotopy perturbation method [13], homotopy analysis method [14, 15], variational iteration method [16, 17], Bäcklund transformation [18, 19], Exp-function method [17, 20, 21], (G′/G)-expansion method [22, 23], Laplace Adomian decomposition method [24], and differential transform method [25]. Here, we use an effective method for constructing a range of exact solutions for following nonlinear partial differential equations that were first proposed by Wang et al. [26] which a new method called the (G′/G)-expansion method to look for travelling wave solutions of NLEEs. Zhang et al. [27] examined the generalized (G′/G)-expansion method and its applications. Authors of [28] used mKdV equation with variable coefficients using the (G′/G)-expansion method. Also, Bekir [23] used application of the (G′/G)-expansion method for nonlinear evolution equations. In this paper we explain the method which is called the (G′/G)-expansion method to look for travelling wave solutions of nonlinear evolution equations.

The paper is organized as follows. In Section 2, we briefly give the steps of the methods and apply the methods to solve the nonlinear partial differential equations. In Section 3 the application of generalized tanh-coth method to the DSSH equation will be introduced briefly, respectively. Also a conclusion is given in Section 4. Finally some references are given at the end of this paper.

3. Application of the Generalized Tanh-Coth Method
We next consider DSSH equation with the generalized tanh-coth method in the following form:
(19)wt-6wwx+wxxx-6vx=0,vt-2vxxx+6wvx=0.
Proceeding as before we get
(20)2u(v)-18u′u′′′-9(u′′)2+12(u′)3-c2u′-cu′′′=0.
In order to determine value of m, we balance u(v) with u′u′′′ in (20), and by using (10) we obtain m=1. We can suppose that the solutions of (19) are of the following form:
(21)u(ξ)=a0+a1Φ+b1Φ.
Substituting (21) into (20) and by using the well-known software Maple, we obtain the system of the following results:
(22)a0=2q, a1=2q, b1=0, c=p2-4qr,
or
(23)a0=2r, a1=0, b1=-2r, c=p2-4qr,
where p, q, r, and c are arbitrary constants. Substituting (22) and (23) into expression (21) we obtain
(24)u(ξ)=2q+2qΦ(ξ), ξ=x-(p2-4qr)t,(25)u(ξ)=2r-2rΦ(ξ), ξ=x-(p2-4qr)t.
By the manipulation as explained in the previous section, we have the following.

(I) The First Set for (24). By using Case 1 from Section 2 we have
(26)u1(x,t)=2q{1-12q[p+Δtanh(Δξ2)]},u2(x,t)=2q{1-12q[p+Δcoth(Δξ2)]},u3(x,t) =2q{1-12q[p+Δ[tanh(Δξ)±i sech(Δξ)]]},u4(x,t) =2q{1-12q[p+Δ[coth(Δξ)±csch(Δξ)]]},u5(x,t) =2q{1-14q0000l00l×[2p+Δ[tanh(Δξ4)00000000000000000l±coth(Δξ4)]]},u6(x,t)=2q{1+12q[-p+(A2+B2)Δ-AΔcosh(Δξ)Asinh(Δξ)+B]},u7(x,t)=2q{1+12q[-p-(B2-A2)Δ+AΔcosh(Δξ)Asinh(Δξ)+B]},u8(x,t) =2q{1+2rcosh(Δξ/2)Δsinh(Δξ/2)-pcosh(Δξ/2)},u9(x,t) =2q{1+-2rsinh(Δξ/2)psinh(Δξ/2)-Δcosh(Δξ/2)},u10(x,t) =2q{1+2rcosh(Δξ/2)Δsinh(Δξ)-pcosh(Δξ)±iΔ},u11(x,t) =2q{1+2rsinh(Δξ/2)-psinh(Δξ)+Δcosh(Δξ)±Δ},u12(x,t)=2q{+2Δcosh2(Δξ/4)-Δ)-1)1+(4rsinh(Δξ/4)cosh(Δξ/4))000000000000l×(-2psinh(Δξ/4)cosh(Δξ/4)00000000000000000+2Δcosh2(Δξ/4)-Δ)-1}.
By using Case 2 from Section 2 we have
(27)u13(x,t)=2q{1-12q[p--Δtan(-Δξ2)]},u14(x,t)=2q{1-12q[p+-Δ cot(-Δξ2)]},u15(x,t) =2q{1-12q[p--Δ[tan(-Δξ)±sec(-Δξ)]]},u16(x,t) =2q{1-12q[p+-Δ[cot(-Δξ)±csc(-Δξ)]]},u17(x,t) =2q{1-14q000l000×[2p--Δ[tan(-Δξ4)000000000000000000l-cot(-Δξ4)]]},u18(x,t) =2q{1+12q000l000×[1+12q-p+(±(A2+B2)Δ-A-Δcos(-Δξ))0000000000 ×(Asin(-Δξ)+B)-11+12q]},u19(x,t)=2q{1+12q000l00×[1+12q-p-(±(B2-A2)Δ+A-Δcos(-Δξ))000000000×(Asin(-Δξ)+B)-11+12q]},u20(x,t) =2q{1+-2rcos(-Δξ/2)-Δsin(-Δξ/2)+pcos(-Δξ/2)},u21(x,t) =2q{1+2rsin(-Δξ/2)-psin(-Δξ/2)+-Δcos(-Δξ/2)},u22(x,t) =2q{1+-2rcos(-Δξ/2)-Δsin(-Δξ)+pcos(-Δξ)±-Δ},u23(x,t)=2q{1+2rsin(-Δξ/2)-psin(-Δξ)+-Δcos(-Δξ)±-Δ},u24(x,t) =2q{+2-Δ cos2(-Δξ4)--Δ)-11+(4rsin(-Δξ4)cosh(-Δξ4))0000000l0000×(-2psin(-Δξ4)cos(-Δξ4)000000000000+2-Δ cos2(-Δξ4)--Δ)-1}.
By using Case 3 from Section 2 we have
(28)u25(x,t)=2q{1+-pdq[d+cosh(pξ)-sinh(pξ)]},u26(x,t)=2q{1+p[cosh(pξ)+sinh(pξ)]q[d+cosh(pξ)+sinh(pξ)]}.
By using Case 4 from Section 2 we have
(29)u27(x,t)=2q{1-1qξ+c},
where ξ=x-(p2-4qr)t.

(II) The Second Set for (25). By using Case 1 from Section 2 we have
(30) u1(x,t)=2r{1+2q[p+Δtanh(Δξ/2)]}, u2(x,t)=2r{1+2q[p+Δcoth(Δξ/2)]},u3(x,t) =2r{1+2q[p+Δ[tanh(Δξ)±i sech(Δξ)]]},u4(x,t) =2r{1+2q[p+Δ[coth(Δξ)±csch(Δξ)]]},u5(x,t) =2r{±coth(Δξ4)]]-11+4q000000l×[2p+Δ[tanh(Δξ4)000000000000000000±coth(Δξ4)]]-1},u6(x,t)=2r{[-p+(((A2+B2)Δ-AΔcosh(Δξ))1-2q000l00 ×[-p+(((A2+B2)Δ-AΔcosh(Δξ))0000000000000000+(Asinh(Δξ)+B)-1)-p+(((A2+B2)Δ-AΔcosh(Δξ))]-1},u7(x,t)=2r{[(Asinh(Δξ)+B)-1]-11-2q000l00×[-p-((B2-A2)Δ+AΔcosh(Δξ))000000000×(Asinh(Δξ)+B)-1]-1},u8(x,t) =2r{1-Δsinh(Δξ/2)-pcosh(Δξ/2)2rcosh(Δξ/2)},u9(x,t) =2r{1+psinh(Δξ/2)-Δcosh(Δξ/2)2rsinh(Δξ/2)},u10(x,t) =2r{1-Δsinh(Δξ)-pcosh(Δξ)±iΔ2rcosh(Δξ/2)},u11(x,t) =2r{1--psinh(Δξ)+Δcosh(Δξ)±Δ2rsinh(Δξ/2)},u12(x,t)=2r{(4rsinh(Δξ4)cosh(Δξ4))-11-(-2psinh(Δξ4)cosh(Δξ4)000000l0000+2Δ cosh2(Δξ4)-Δ)0000000×(4rsinh(Δξ4)cosh(Δξ4))-1}.
By using Case 2 from Section 2 we have
(31)u13(x,t)=2r{1+2q[p--Δtan(-Δξ/2)]},u14(x,t)=2r{1+2q[p+-Δ cot(-Δξ/2)]},u15(x,t) =2r{1+2q[p--Δ[tan(-Δξ)±sec(-Δξ)]]},u16(x,t) =2r{1+2q[p+-Δ[cot(-Δξ)±csc(-Δξ)]]},u17(x,t)=2r{1+4q[2p--Δ[tan(-Δξ/4)-cot(-Δξ/4)]]},u18(x,t)=2r{(Asin(-Δξ)+B)-1]-11-2q000l00×[-p+(±(A2+B2)Δ-A-Δcos(-Δξ))000000000×(Asin(-Δξ)+B)-1]-1},u19(x,t)=2r{×(Asin(-Δξ)+B)-1]-11-2q00l00×[-p-(±(B2-A2)Δ+A-Δcos(-Δξ))00000000×(Asin(-Δξ)+B)-1]-1},u20(x,t) =2r{1+-Δsin(-Δξ/2)+pcos(-Δξ/2)2rcos(-Δξ/2)},u21(x,t) =2r{1--psin(-Δξ/2)+-Δcos(-Δξ/2)2rsin(-Δξ/2)},u22(x,t) =2r{1+-Δsin(-Δξ)+pcos(-Δξ)±-Δ2rcos(-Δξ/2)},u23(x,t) =2r{1--psin(-Δξ)+-Δcos(-Δξ)±-Δ2rsin(-Δξ/2)},u24(x,t) =2r{(4rsin(-Δξ4)cosh(-Δξ4))-11-(-2psin(-Δξ4)cos(-Δξ4)000000l0000+2-Δcos2(-Δξ4)--Δ)0000000×(4rsin(-Δξ4)cosh(-Δξ4))-1}.
By using Case 3 from Section 2 we have
(32)u25(x,t)=2r{1+q[d+cosh(pξ)-sinh(pξ)]pd},u26(x,t)=2r{1-q[d+cosh(pξ)+sinh(pξ)]p[cosh(pξ)+sinh(pξ)]}.
By using Case 4 from Section 2 we have
(33)u27(x,t)=2r{1+qξ+c},
where ξ=x-(p2-4qr)t. By using the w(x,t)=ux(x,t), v(x,t)=(1/6)(ut-3(ux)2+uxxx), can be used to get the solutions of the DSSH system (19). It can be seen that the results are the same, with comparing results in the literature [6]. We obtained analytical solutions by the generalized (G′/G)-expansion and the generalized tanh-coth methods. Also, in this paper we can see correlation between (G′/G)-expansion method and tanh-coth methods. We have succeeded in identifying the equivalence of the two methods under special conditions [33]. Consequently, the solution of the equations via (G′/G)-expansion method is exactly the same as the solution of tanh-coth method if the conditions are satisfied. In fact, we can see that the tanh-coth method is a special case of the (G′/G)-expansion method.