1. Introduction
In this paper, we consider the following double nonlinear dispersive, integrable equation:
(1)ut+α(u3)x+32α(u2)xx+αuxxx=0,
where α is a real parameter and u(x,t) is the unknown function depending on the temporal variable t and the spatial variable x. This equation contains both linear dispersive term αuxxx and the double nonlinear terms α(u2)xx and α(u3)x. Equation (1) be called SharmaTassoOlver equation in the literature. Many physicists and mathematicians have paied their attentions to the SharmaTassoOlver equation in recent years due to its appearance in scientific applications. In [1], the tanh method, the extended tanh method, and other ansatz involving hyperbolic and exponential functions are efficiently used for the analytic study of this equation. The multiple solitons and kinks solutions are obtained. In [2], Yan investigated the SharmaTassoOlver equation (1) by using the ColeHopf transformation method. The simple symmetry reduction procedure is repeatedly used in [3] to obtain exact solutions where soliton fission and fusion were examined. Wang et al. examined the soliton fission and fusion thoroughly by means of the Hirotas bilinear method and the Bäcklund transformation method in [4]. The generalized KaupNewelltype hierarchy of nonlinear evolution equations is explicitly related to SharmaTassoOlver equation from [5]. Using the improved tanh function method in [6], the SharmaTassoOlver equation with its fission and fusion has some exact solutions. In [7], some exact solution of the SharmaTassoOlver equation is given by implying a generalized tanh function method for approximating some solutions which have been known.
In recent years, with the development of symbolic computation packages like Maple and Mathematica, which enable us to perform the tedious and complex computation on computer, much work has been focused on the direct methods to construct exact solutions of nonlinear evolution equations. The G′/Gexpansion method proposed by Wang et al. [8] is one of the most effective direct methods to obtain travelling wave solutions of a large number of nonlinear evolution equations, such as the KdV equation, the mKdV equation, the variant Boussinesq equations, and the HirotaSatsuma equations. Later, the further developed methods, named, the generalized G′/Gexpansion method, the modified G′/Gexpansion method, and the extended G′/Gexpansion method have been proposed in [9–11], respectively. The aim of this paper is to derive more exact solitary wave solutions and periodic wave solutions of the SharmaTassoOlver equation. We will employ the improved G′/G method to solve SharmaTassoOlver equation. Some entirely new exact solitary wave solutions and periodic wave solutions of the SharmaTassoOlver equation are obtained.
The rest of the paper is organized as follows. In Section 2, we describe the method in brief. In Sections 3, we study the SharmaTassoOlver equation by the improved G′/Gexpansion method. Finally, we give the conclusion.
2. The Improved Method
The main steps of improved G′/Gexpansion method [12, 13] are introduced as follows.
Step 1.
Consider a general nonlinear PDE in the form
(2)F(u,ux,ut,uxx,uxt,…)=0.
Using a wave variable
(3)u(x,t)=U(ξ), ξ=xct,
we can rewrite (2) as the following nonlinear ODE:
(4)F(U,U′,U′′,…)=0,
where the prime denotes differentiation with respect to ξ.
Step 2.
Suppose that the solution of ODE (4) can be written as follows:
(5)U(ξ)=∑i=nnai(G′G+σG′)i,
where σ,ai (i=n,n+1,…) are constants to be determined later, n is a positive integer, and G=G(ξ) satisfies the following secondorder linear ordinary differential equation:
(6)G′′+λG′+μG=0,
where λ,μ is a real constant. The general solutions of (6) can be listed as follows. When ▵=λ24μ>0, we obtain the hyperbolic function solution of (6)
(7)G(ξ)=e(λ/2)ξ(A1cosh(▵2ξ)+A2sinh(▵2ξ)),
where A1 and A2 are arbitrary constants. When ▵=λ24μ<0, we obtain the following trigonometric function solution of (6):
(8)G(ξ)=e(λ/2)ξ(A1cos(▵2ξ)+A2sin(▵2ξ)),
where A1 and A2 are arbitrary constants. When ▵=λ24μ=0, we obtain the solution of (6) as
(9)G(ξ)=e(λ/2)ξ(A1+A2ξ),
where A1 and A2 are arbitrary constants.
Step 3.
Determine the positive integer n by balancing the highest order derivatives and nonlinear terms in (4).
Step 4.
Substituting (5) along with (6) into (4) and then setting all the coefficients of (G′/G)k (k=1,2,…) of the resulting system’s numerator to zero yield a set of overdetermined nonlinear algebraic equations for c,σ, and ai (i=n,n+1,…).
Step 5.
Assuming that the constants c,σ, and ai (i=n,n+1,…) can be obtained by solving the algebraic equations in Step 4, then substituting these constants and the known general solutions of (6) into (5), we can obtain the explicit solutions of (2) immediately.
3. The Exact Solutions of the SharmaTassoOlver Equation
In this section, we will construct travelling wave solutions of the SharmaTassoOlver equation (1) by using the method described in Section 2.
Let u(x,t)=φ(xct)=φ(ξ), where c is the wave speed. Substituting the above travelling wave variable ξ=xct into SharmaTassoOlver equation (1) yields
(10)cφ′+α(φ3)′+32α(φ2)′′+αφ′′′=0.
By integrating (10) with respect to the variable ξ and assuming a zero constant of integration, we obtain the following nonlinear ordinary differential equation for the function φ:
(11)cφ+αφ3+3αφφ′+αφ′′=0.
Balancing φ′′ with φ3 in (11), we find n+2=3n so that n=1, and we suppose that (11) owns the solutions in the form
(12)φ(ξ)=a0+a1G′G+σG′+b1(G′G+σG′)1.
Substituting (12) along with (6) into (11) and then setting all the coefficients of (G′/G)k (k=0,1,…) of the resulting system’s numerator to zero yield a set of overdetermined nonlinear algebraic equations about a0,a1,b1,c. Solving the overdetermined algebraic equations, we can obtain the following results.
Case 1.
(13)
a
0
=
μ
σ

λ
2
,
a
1
=
σ
λ

σ
2
μ

1
,
b
1
=
0
,
c
=

1
4
(
4
μ

λ
2
)
α
,
where σ are arbitrary constants.
Case 2.
(14)
a
0
=

λ
+
2
μ
σ
,
a
1
=
2
σ
λ

2
σ
2
μ

2
,
b
1
=
0
,
c
=

(
4
μ

λ
2
)
α
,
where σ are arbitrary constants.
Case 3.
(15)
a
0
=
μ
σ

λ
2
±
λ
2

4
μ
2
,
a
1
=
σ
λ

σ
2
μ

1
,
b
1
=
0
,
c
=

(
4
μ

λ
2
)
α
,
where σ are arbitrary constants.
Case 4.
(16)
a
0
=

2
μ
σ
+
λ
,
a
1
=
0
,
b
1
=
2
μ
,
c
=
α
λ
2

4
α
μ
,
where σ are arbitrary constants.
Case 5.
(17)
a
0
=
0
,
a
1
=
σ
λ

σ
2
μ

1
,
b
1
=
μ
,
c
=
α
λ
2

4
α
μ
,
where σ are arbitrary constants.
Case 6.
(18)
a
0
=

μ
σ
+
λ
2
,
a
1
=
0
,
b
1
=
μ
,
c
=
1
4
α
λ
2

α
μ
,
where σ are arbitrary constants.
Case 7.
(19)
a
0
=
μ
σ

λ
2
±
λ
2

4
μ
2
,
a
1
=
0
,
b
1
=
μ
,
c
=

(
4
μ

λ
2
)
α
,
where σ are arbitrary constants.
Using Case 5, (12), and the general solutions of (6), we can find the following travelling wave solutions of SharmaTassoOlver equation (1).
When ▵=λ24 μ>0, we obtain the hyperbolic function solutions of (1) as follows:
(20)u(x,t)=φ(ξ)=a1G′G+σG′+b1(G′G+σG′)1=(σλσ2μ1) ×({+(2A2σλA2+σA1▵)sinh(▵2ξ))1}((λA1+A2▵)cosh(▵2ξ) +(λA2+A1▵)sinh(▵2ξ)) ×((2A1σλA1+σA2▵)cosh(▵2ξ) +(2A2σλA2+σA1▵) ×sinh(▵2ξ))1) +μ({+(λA2+A1▵)sinh(▵2ξ))1}((2A1σλA1+σA2▵)cosh(▵2ξ) +(2A2σλA2+σA1▵) ×sinh(▵2ξ)) ×((λA1+A2▵)cosh(▵2ξ) +(λA2+A1▵) ×sinh(▵2ξ))1),
where ξ=xct, c=αλ24αμ, A1,A2,σ are arbitrary constants.
It is easy to see that the hyperbolic function solution can be rewritten at A12<A22 and A12>A22 as follows:(21a)u(x,t)=φ(ξ)=(σλσ2μ1)×λ+λ24μtanh((λ24μ/2)ξ+ξ0)2σλ+σλ24μtanh((λ24μ/2)ξ+ξ0)+μ2σλ+σλ24μtanh((λ24μ/2)ξ+ξ0)λ+λ24μtanh((λ24μ/2)ξ+ξ0),(21b)u(x,t)=φ(ξ)=(σλσ2μ1)×λ+λ24μcoth((λ24μ/2)ξ+ξ0)2σλ+σλ24μcoth((λ24μ/2)ξ+ξ0)+μ2σλ+σλ24μcoth((λ24μ/2)ξ+ξ0)λ+λ24μcoth((λ24μ/2)ξ+ξ0),
where ξ=xct, c=4αμ, and ξ0=tanh1(A2/A1).
Specially, if σ=0, (21a) and (21b) become(22a)u(x,t)=φ(ξ)=λ2 λ24μ2tanh( λ24μ2ξ+ξ0)+ 2μλ+λ24μtanh((λ24μ/2)ξ+ξ0),(22b)u(x,t)=φ(ξ)=λ2 λ24μ2coth(λ24μ2ξ+ξ0)+ 2μλ+λ24μcoth((λ24μ/2)ξ+ξ0),
where ξ=xct, c=αλ24αμ, and ξ0=tanh1(A2/A1).
Taking λ=0 in (22a) and (22b), we have
(23)u(x,t)=φ(ξ)=±(μ coth(μξ+ξ0)μtanh(μξ+ξ0)),
where ξ=xct, c=4αμ, and ξ0=tanh1(A2/A1), μ>0.
It is easy to see that if A1,A2,σ,λ,μ are taken as other special values in a proper way, more solitary wave solutions of (1) can be obtained. Here we omit them for simplicity.
When ▵=λ24μ<0, we get the trigonometric function solutions of (1) as follows:
(24)u(x,t)=φ(ξ)=a1G′G+σG′+b1(G′G+σG′)1=(σλσ2μ1)×(((λA1+A2▵)cos(▵2ξ) +(λA2A1▵)sin(▵2 ξ)) ×({(▵2ξ)}(2A1σλA1σA2▵) ×cos(▵2ξ) +(2A2σ λA2+σA1▵) ×sin(▵2ξ))1)+μ({+(λA2A1▵)sin(▵2 ξ))1}({(▵2ξ)}(2A1σλA1σA2▵) ×cos(▵2ξ) +(2A2σλA2+σA1▵) ×sin(▵2ξ)) ×((λA1+A2▵)cos(▵2ξ) +(λA2A1▵) ×sin(▵2 ξ))1),
where ξ=xct, c=αλ24αμ, A1,A2,σ are arbitrary constants.
It is easy to see that the trigonometric solution can be rewritten at A12<A22 and A12>A22 as follows:(25a)u(x,t) =φ(ξ) =(σλσ2μ1) ×λ+λ2+4μtan((λ2+4μ/2)ξ+ξ0)2σλ+σλ2+4μtan((λ2+4μ/2)ξ+ξ0) +μ2σλ+σλ2+4μtan((λ2+4μ/2)ξ+ξ0)λ+λ2+4μtan((λ2+4μ/2)ξ+ξ0),(25b)u(x,t) =φ(ξ) =(σλσ2μ1) ×λ+λ2+4μ cot((λ2+4μ/2)ξ+ξ0)2σλ+σλ2+4μ cot((λ2+4μ/2)ξ+ξ0) +μ2σλ+σλ2+4μ cot((λ2+4μ/2)ξ+ξ0)λ+λ2+4μcot((λ2+4μ/2)ξ+ξ0),
where ξ=xct, c=4αμ, and ξ0=tan1(A2/A1).
Specially, if σ=0, (25a) and (25b) become(26a)u(x,t)=φ(ξ)=λ2λ2+4μ2tan(λ2+4μ2ξ+ξ0)+2μλ+λ2+4μtan((λ2+4μ/2)ξ+ξ0),(26b)u(x,t)=φ(ξ)=λ2λ2+4μ2cot(λ2+4μ2ξ+ξ0)+2μλ+λ2+4μ cot((λ2+4μ/2)ξ+ξ0),
where ξ=xct, c=αλ24αμ, and ξ0=tan1(A2/A1).
Taking λ=0 in (26a) and (26b), we have
(27)u(x,t)=φ(ξ)=±(μ cot(μξ+ξ0)μtan(μξ+ξ0)),
where ξ=xct, c=4αμ, and ξ0=tan1(A2/A1), μ>0.
Using other 6 cases, (12) and the general solutions of (6), we could obtain abundant exact solutions of (1), and here we do not list all of them.