Turk J Phys 34 (2010) , 73 – 82. ¨ ITAK ˙ c TUB doi:10.3906/fiz-0909-7
Exact solutions for the double sinh-Gordon and generalized form of the double sinh-Gordon equations by using (G /G)-expansion method Hossein KHEIRI and Azizeh JABBARI Faculty of Mathematical Sciences, University of Tabriz, Tabriz-IRAN e-mails:
[email protected],
[email protected]
Received 06.09.2009
Abstract In this paper, the (G /G) -expansion method is applied to seek traveling wave solutions to the double sinh-Gordon and the generalized form of the double sinh-Gordon equations. With the aid of a symbolic computation system, two types of more general traveling wave solutions (including hyperbolic functions and trigonometric functions) with free parameters are constructed. Solutions concerning solitary and periodic waves are also given by setting the two arbitrary parameters, involved in the traveling waves, as special values. Key Words: (G /G) -expansion method, double sinh-Gordon equation, generalized form of the double sinh-Gordon equation, traveling wave solutions PACS: 02.30.Jr; 05.45.Yv
1.
Introduction The sinh-Gordon equation, utt − uxx + sinh u = 0,
(1)
gained its importance because of the kink and antikink solutions with the collisional behaviors of solitons that arise from this equation. The equation appears in integrable quantum field theory, kink dynamics, and fluid dynamics [1–8]. Many powerful methods, such as B a ¨ cklund transformation, inverse scattering method, Hirota bilinear forms, pseudo spectral method, the tanh method, tanh-sech method, the sine-cosine method [9–14], and many others were successfully applied to nonlinear equations. Recently, Wang et al. [15] proposed the (G /G)expansion method and showed that it is powerful for finding analytic solutions of PDEs. Next, Bekir [16] 73
KHEIRI, JABBARI
applied the method to some nonlinear evolution equations gaining traveling wave solutions. More recently,
Zhang et al. [17] proposed a generalized ( G G )-expansion method to improve and extend Wang et al.’s work [15] for solving variable coefficient equations and high dimensional equations. Kheiri et al. applied this method for solving the combined and the double combined sinh-cosh-Gordon equations [18]. Also, Zhang [17] solved the equations with the balance numbers of which are not positive integers, by this method.
The ( GG )-expansion method is based on the explicit linearization of nonlinear differential equations for traveling waves with a certain substitution which leads to a second-order differential equation with constant coefficients. Computations are performed with a computer algebra system such as Maple to deduce the solutions of the nonlinear equations in an explicit form. The solution process of the method is direct, effective and convenient due to solving the auxiliary equation of second-order differential equation with constant coefficients. We should mention that this technique is restricted to the search for single soliton solutions. To formally derive N-soliton solutions of any completely integrable equation, authors mainly used the Cole-Hopf transformation method combined with the Hirota’s bilinear method, where it was shown that soliton solutions are just polynomials of exponentials [19]. In this paper we apply the (G /G)-expansion method to the double sinh-Gordon equation and its generalized form.
2.
Description of the (G /G)-expansion method We suppose that the given nonlinear partial differential equation for u(x, t) has the form P (u, ux, ut , uxx, uxt , utt, ...) = 0,
(2)
where P is a polynomial in its arguments. The essence of the (G /G)-expansion method is the following steps: Step 1. Seek traveling wave solutions of equation (2) by taking u(x, t) = U (ξ), ξ = x − ct, and transform equation (2) to the ordinary differential equation Q(U, U , U , ...) = 0,
(3)
where prime denotes the derivative with respect to ξ . Step 2. If possible, integrate equation (3) term by term one or more times. This yields constant(s) of integration. For simplicity, the integration constant(s) can be set to zero. Step 3. Introduce the solution U (ξ) of equation (3) in the finite series form U (ξ) =
N i=0
ai
G (ξ) G(ξ)
i ,
(4)
where ai are real constants with aN = 0 to be determined, N is a positive integer to be determined. The function G(ξ) is the solution of the auxiliary linear ordinary differential equation G (ξ) + λG (ξ) + μG(ξ) = 0,
(5)
where λ and μ are real constants to be determined. Step 4. Determine N . This, usually, can be accomplished by balancing the linear term(s) of highest order 74
KHEIRI, JABBARI
with the highest order nonlinear term(s) in equation (3). Step 5. Substituting (4) together with (5) into equation (3) yields an algebraic equation involving powers of (G /G). Equating the coefficients of each power of (G /G) to zero gives a system of algebraic equations for ai , λ, μ and c. Then, we solve the system with the aid of a computer algebra system, such as Maple, to determine these constants. On the other hand, depending on the sign of the discriminant Δ = λ2 − 4μ, the solutions of equation (5) are well known to us. So, as a final step, we can obtain exact solutions of equation (2).
3.
The double sinh-Gordon equation We first solve the double sinh-Gordon equation utt − kuxx + 2α sinh u + β sinh(2u) = 0,
(6)
where α and β are nonzero real constants. Making the transformation u(x, t) = u(ξ), ξ = x−ct and integrating once with respect to ξ , we get (c2 − k)u + 2α sinh u + β sinh(2u) = 0.
(7)
By applying the Painlev´e transformation, v = eu ,
(8)
u = ln v,
(9)
or equivalently,
we have sinh(u) =
v − v−1 , 2
sinh(2u) =
then
v2 − v−2 v + v−1 , cosh(u) = , 2 2
u = arccosh
v + v−1 . 2
(10)
(11)
Consequently, we can write the double sinh-Gordon equation (7) as the ODE βv4 + 2αv3 − 2αv − β + 2(c2 − k)vv − 2(c2 − k)(v )2 = 0.
(12)
Balancing v4 with vv gives M = 1.
(13)
The (G /G)-expansion method allows us to use the finite expansion v(ξ) = a0 + a1
G G
,
a1 = 0.
(14)
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KHEIRI, JABBARI
Substituting (14) into (12), setting coefficients of ( GG )i (i = 0, 1, ..., 4) to zero, we obtain the following underdetermined system of algebraic equations for a0 , a1 , c, λ and μ:
G G G G
G G G G G G
0 : −β + βa0 4 + 2αa0 3 − 2αa0 − 2c2 a1 2 μ2 + 2ka1 2 μ2 + 2c2 a0 a1 λμ − 2ka0 a1 λμ = 0, 1 : −2αa1 + 4βa1 a0 3 + 6αa1 a0 2 + 2c2 a0 a1 λ2 + 4c2 a0 a1 μ − 2c2 a1 2 λμ − 2ka0 a1 λ2
2
−4ka0 a1 μ + 2ka1 2 λμ = 0, : 6βa1 2 a0 2 + 6αa1 2 a0 + 6c2 a0 a1 λ − 6ka0 a1 λ = 0,
3 : 2αa1 3 + 4βa1 3 a0 + 4c2 a0 a1 + 2c2 a1 2 λ − 4ka0 a1 − 2ka1 2 λ = 0, 4 : βa1 4 + 2c2 a1 2 − 2ka1 2 = 0.
Solving this system using Maple gives a0
=
a1
=
c =
−α λ α2 − β 2 ± , β β λ2 − 4μ 2 α2 − β 2 ± , β λ2 − 4μ α2 − β 2 ± k−2 , β(λ2 − 4μ)
α > β, c2 < k,
(15)
where λ and μ are arbitrary constants. Substituting equation (15) into equation (14) yields v(ξ)
=
−α λ ± β β
α2 − β 2 2 ± λ2 − 4μ β
α2 − β 2 G ( ), λ2 − 4μ G
(16)
α2 −β 2 t. where ξ = x − ± k − 2 β(λ 2 −4μ) Substituting general solutions of equation (5) into equation (16), we have two types of traveling wave solutions of the double sinh-Gordon equation as follows. • When λ2 − 4μ > 0 , v1 (ξ)
=
−α ± β
α2 − β 2 β
α2 −β 2 t. where ξ = x − ± k − 2 β(λ 2 −4μ)
76
λ2 − 4μξ + c2 cosh 12 λ2 − 4μξ , α > β, c1 cosh 12 λ2 − 4μξ + c2 sinh 12 λ2 − 4μξ
c1 sinh 12
(17)
KHEIRI, JABBARI
• When λ2 − 4μ < 0 ,
v2 (ξ) =
−α ± β
β 2 − α2 −c1 sin 12 4μ − λ2 ξ + c2 cos 12 4μ − λ2 ξ , α < β, β c1 cos 12 4μ − λ2 ξ + c2 sin 12 4μ − λ2 ξ
(18)
α2 −β 2 t. where ξ = x − ± k − 2 β(λ 2 −4μ) Recall that
v + v−1 u = arccosh . 2
(19)
Therefore, we obtain the solutions, for α > β , k > c2 , u1 (ξ)
α2 − β 2 c1 sinh 12 λ2 − 4μξ + c2 cosh 12 λ2 − 4μξ 1 −α = arccosh ± 2 β β c1 cosh 12 λ2 − 4μξ + c2 sinh 12 λ2 − 4μξ
−1 ⎞⎫ ⎬ 1 1 2 2 2 2 c1 sinh 2 λ − 4μξ + c2 cosh 2 λ − 4μξ α −β −α ⎠ , ± + ⎭ β β c1 cosh 1 λ2 − 4μξ + c2 sinh 1 λ2 − 4μξ 2
(20)
2
for α < β , k > c2 , u2 (ξ)
β 2 − α2 −c1 sin 12 4μ − λ2 ξ + c2 cos 12 4μ − λ2 ξ 1 −α = arccosh ± 2 β β c1 cos 12 4μ − λ2 ξ + c2 sin 12 4μ − λ2 ξ
−1 ⎞⎫ ⎬ β 2 − α2 −c1 sin 12 4μ − λ2 ξ + c2 cos 12 4μ − λ2 ξ −α ⎠ , ± + ⎭ β β c1 cos 1 4μ − λ2 ξ + c2 sin 1 4μ − λ2 ξ 2
(21)
2
α2 −β 2 t. In solutions (20) and (21), c1 and c2 are left as free parameters. where ξ = x − ± k − 2 β(λ 2 −4μ) In particular, if c1 = 0 , c2 = 0 , then u1 becomes u1 (ξ)
=
α2 − β 2 α2 − β 2 1 −α arccosh ± tanh (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ α2 − β 2 α2 − β 2 −α ⎠ , ± tanh (x − ct) + ⎭ β β 2β(k − c2 )
(22)
if c2 = 0 , c1 = 0 , u1 (ξ)
=
α2 − β 2 α2 − β 2 1 −α arccosh ± coth (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ 2 2 2 2 α −β α −β −α ⎠ , ± coth (x − ct) + ⎭ β β 2β(k − c2 )
(23)
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where α > β and k > c2 , which are the solitary solutions of the double sinh-Gordon equation. If c1 = 0 , c2 = 0 , u2 becomes u2 (ξ)
=
β 2 − α2 β 2 − α2 1 −α arccosh ∓ tan (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ β 2 − α2 β 2 − α2 α ⎠ , ± tan (x − ct) − ⎭ β β 2β(k − c2 )
(24)
if c2 = 0 , c1 = 0 , u2 (ξ)
β 2 − α2 β 2 − α2 1 −α = arccosh ± cot (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ β 2 − α2 β 2 − α2 α ⎠ , ∓ cot (x − ct) − ⎭ β β 2β(k − c2 )
(25)
where β > α and k > c2 . These are the periodic solutions of the double sinh-Gordon equation. These special results show that our method obtains general solutions and it can be seen that the solutions obtained by using tanh function method [20, 21], are special cases of our solutions.
4.
Generalized form of the double sinh-Gordon equation In this section we consider generalized form of the double sinh-Gordon equation [8] given by utt − kuxx + 2α sinh(nu) + β sinh(2nu) = 0, n ≥ 1.
(26)
Here, α and β are nonzero real constants. Making the transformation u(x, t) = u(ξ), ξ = x−ct, and integrating once with respect to ξ , we get (c2 − k)u + 2α sinh(nu) + β sinh(2nu) = 0.
(27)
Proceeding as before, we use the transformation, v = enu ,
(28)
or equivalently u=
1 ln v, n
(29)
we have sinh(nu) = then
78
v − v−1 , 2
sinh(2nu) =
v2 − v−2 v + v−1 , cosh(nu) = , 2 2
v + v−1 1 . u = arccosh n 2
(30)
(31)
KHEIRI, JABBARI
As a result, this transformation will change the generalized form of double sinh-Gordon equation (27) to the ODE βnv4 + 2αnv3 − 2αnv − βn + 2(c2 − k)vv − 2(c2 − k)(v )2 = 0. (32) Balancing the v4 with vv gives M = 1. Consequently, we have
v(ξ) = a0 + a1
G G
(33)
,
a1 = 0.
(34)
Substituting (34) into (32), setting coefficients of ( GG )i (i = 0, 1, ..., 4) to zero, we obtain the following underdetermined system of algebraic equations for a0 , a1 , c, λ and μ:
G G G G
0 : −βn + βna0 4 + 2αna0 3 − 2αna0 − 2c2 a1 2 μ2 + 2ka1 2 μ2 + 2c2 a0 a1 λμ − 2ka0 a1 λμ = 0, 1 : −2αna1 + 4βna1 a0 3 + 6αna1 a0 2 + 2c2 a0 a1 λ2 + 4c2 a0 a1 μ − 2c2 a1 2 λμ − 2ka0 a1 λ2 −4ka0 a1 μ + 2ka1 2 λμ = 0,
G G G G G G
2 : 6βna1 2 a0 2 + 6αna1 2 a0 + 6c2 a0 a1 λ − 6ka0 a1 λ = 0, 3 : 2αna1 3 + 4βna1 3 a0 + 4c2 a0 a1 + 2c2 a1 2 λ − 4ka0 a1 − 2ka1 2 λ = 0, 4 : βna1 4 + 2c2 a1 2 − 2ka1 2 = 0.
Solving this system by Maple, gives
a0
=
a1
=
c =
−α λ α2 − β 2 ± , β β λ2 − 4μ 2 α2 − β 2 ± , β λ2 − 4μ α2 − β 2 ± k − 2n , β(λ2 − 4μ)
α > β, c2 < k,
(35)
where λ and μ are arbitrary constants. Substituting equation (35) into equation (34) yields
v(ξ)
=
−α λ ± β β
α2 − β 2 2 ± λ2 − 4μ β
α2 − β 2 λ2 − 4μ
G G
,
(36)
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α2 −β 2 t. where ξ = x − ± k − 2n β(λ 2 −4μ) Substituting general solutions of equation (5) into equation (36), we have two types of traveling wave solutions of the generalized form of the double sinh-Gordon equation as follows. • When λ2 − 4μ > 0 ,
v1 (ξ)
=
−α ± β
α2 − β 2 c1 sinh 12 λ2 − 4μξ + c2 cosh 12 λ2 − 4μξ ( ), α > β, β c1 cosh 1 λ2 − 4μξ + c2 sinh 1 λ2 − 4μξ 2
(37)
2
α2−β 2 where ξ = x − (± k − 2n β(λ 2 −4μ) )t. • When λ2 − 4μ < 0 ,
v2 (ξ)
=
−α ± β
β 2 − α2 −c1 sin 12 4μ − λ2 ξ + c2 cos 12 4μ − λ2 ξ ), α < β, ( β c1 cos 1 4μ − λ2 ξ + c2 sin 1 4μ − λ2 ξ 2
(38)
2
α2−β 2 where ξ = x − (± k − 2n β(λ 2 −4μ) )t. Recall that u=
1 v + v−1 arccosh , n 2
(39)
therefore, we obtain the solutions, for α > β , k > c2 ,
u1 (ξ)
=
α2 − β 2 c1 sinh 12 λ2 − 4μξ + c2 cosh 12 λ2 − 4μξ 1 −α ± 2 β β c1 cosh 12 λ2 − 4μξ + c2 sinh 12 λ2 − 4μξ
−1 ⎞⎫ ⎬ α2 − β 2 c1 sinh 12 λ2 − 4μξ + c2 cosh 12 λ2 − 4μξ −α ⎠ , ± + ⎭ β β c1 cosh 1 λ2 − 4μξ + c2 sinh 1 λ2 − 4μξ
1 arccosh n
2
(40)
2
for α < β , k > c2 ,
u2 (ξ)
=
β 2 − α2 −c1 sin 12 4μ − λ2 ξ + c2 cos 12 4μ − λ2 ξ 1 −α ± 2 β β c1 cos 12 4μ − λ2 ξ + c2 sin 12 4μ − λ2 ξ
−1 ⎞⎫ ⎬ 1 1 2 2 2 2 −c1 sin 2 4μ − λ ξ + c2 cos 2 4μ − λ ξ β −α α ⎠ , ∓ − ⎭ β β c1 cos 1 4μ − λ2 ξ + c2 sin 1 4μ − λ2 ξ
1 arccosh n
2
α2 −β 2 t. where ξ = x − ± k − 2n β(λ 2 −4μ) In solutions (40), (41), c1 and c2 are left as free parameters. 80
2
(41)
KHEIRI, JABBARI
In particular, if c1 = 0 , c2 = 0 , then u1 becomes
u1 (ξ)
=
α2 − β 2 n(α2 − β 2 ) 1 −α ± tanh (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ 2 2 2 2 α −β n(α − β ) −α ⎠ . ± tanh (x − ct) + 2 ⎭ β β 2β(k − c )
1 arccosh n
(42)
If c2 = 0 , c1 = 0 ,
u1 (ξ)
=
α2 − β 2 n(α2 − β 2 ) 1 −α ± coth (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ 2 2 2 2 α −β n(α − β ) −α ⎠ , ± coth (x − ct) ⎭ β β 2β(k − c2 )
1 arccosh n
(43)
where α > β and k > c2 . These are the solitary solutions of the generalized form of the double sinh-Gordon equation. If c1 = 0 , c2 = 0 , u2 becomes
u2 (ξ)
=
β 2 − α2 n(β 2 − α2 ) 1 −α ∓ tan (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ 2 2 2 2 β −α n(β − α ) α ⎠ , ± tan (x − ct) − ⎭ β β 2β(k − c2 )
1 arccosh n
(44)
If c2 = 0 , c1 = 0 ,
u2 (ξ)
=
β 2 − α2 n(β 2 − α2 ) 1 −α ± cot (x − ct) 2 β β 2β(k − c2 ) −1 ⎞⎫ ⎬ 2 2 2 2 β −α n(β − α ) α ⎠ , ∓ cot (x − ct) − 2 ⎭ β β 2β(k − c )
1 arccosh n
(45)
where β > α and k > c2 . These are the periodic solutions of the generalized form of the double sinh-Gordon equation. Again, comparing these special results with Wazwaz’s results [20] shows that our results are more general.
5.
Conclusions
In this paper, the ( GG )-expansion method is used to conduct an analytic study on the double sinh-Gordon and generalized form of the double sinh-Gordon equations. The exact traveling wave solutions being determined 81
KHEIRI, JABBARI
in this study are more general, and it is not difficult to arrive at some known analytic solutions for certain choices of the parameters c1 and c2 . Comparing the proposed method with the methods used in [1–8, 15], show that
the ( G G )-expansion method is not only simple and straightforward, but also avoids tedious calculations.
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