Application of conservation theorem and modified extended tanh-function method to (1+1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation

Chaos, Solitons and Fractals 104 (2017) 33–40

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Application of conservation theorem and modified extended tanh-function method to (1+1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation Arzu Akbulut∗, Filiz Tas¸ can ˙ Eskis¸ ehir Osmangazi University, Art-Science Faculty,Department of Mathematics-Computer, Eskis¸ ehir-TÜRKIYE

a r t i c l e

i n f o

Article history: Received 10 May 2017 Revised 17 July 2017 Accepted 31 July 2017

a b s t r a c t We found trivial conservation laws by conservation theorem and exact solutions modified extended tanhfunction method of (1+1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation. The travelling wave solutions are expressed by the hyperbolic, trigonometric and rational functions. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in the science of mathematics, physics and engineering.

Keywords: Conservation laws Exact solutions Partial differential equation

1. Introduction Nonlinear partial differential equations (PDEs) have become useful for science and engineering. Nonlinear partial differential equations are not arised only from many fields of mathematics, but also from different discipline such as fluid mechanics, plasma physics, optical fibers, solid state physics, chemical kinematic, chemical physics and geochemistry. One of the most powerful method for obtaining solutions of nonlinear partial differential equations is based on the study of their invariance under one parameter Lie group of point transformations. This method is used to analyze the symmetries of the differential equations. Then, obtained symmetry groups can be also used to simplify the given differential equations. It is known that conservation laws play an important role in the solutions of differential equations or system of differential equations. All of the conservation laws for PDEs have no physical meanings, but they are necessary for studying the integrability of PDEs [11]. At the mathematical level, conservation laws are highly connected to the existence of a variational principle which enables symmetry transformations. This crucial fact was fully acknowledged by Emmy Noether in 1918 [8]. Studies of consevation laws of PDEs are becoming attractive in nonlinear science day by day. Some matematicians such as Steudel, Anco and Bluman, Wolf, Kara and Mahomed, Ibragimov, Olver have numerous studies in that field [23]. There are lots of ways for con∗

Corresponding author. E-mail addresses: [email protected] (A. Akbulut), [email protected] (F. Tas¸ can). http://dx.doi.org/10.1016/j.chaos.2017.07.025 0960-0779/© 2017 Elsevier Ltd. All rights reserved.

© 2017 Elsevier Ltd. All rights reserved.

struction of conservation laws. For example; characteristic method, variational approach [25], symmetry and conservation law relation [4,18], direct construction method for conservation laws [2], partial Noether approach [24], Noether approach [22], conservation theorem [5,12,14,15]. In the literature, it is mostly studied with partial differential equations containing real valued dependent variables to find conservation laws but in this study we will deal with partial differential equations containing complex valued dependent variables. Therefore we will consider (1+1)-dimensional nonlinear coupled Klein-Gordon-Zakharov equation. We will apply conservation theorem to this equation for finding it’s conservation laws. This method is based on the concept of Lie symmetry generators, adjoint equations and formal Lagrangians. We will deal with the exact solutions after finding the conservation laws of the given equation. Exact traveling wave solutions of nonlinear PDEs are important for understanding qualitative property of many various areas of natural science. For inctance, exact solutions let researchers to design and operate tests by creating suitable conditions and determine these parameters or functions. Recently, many new methods have been proposed for finding these solutions, such as Jacobi elliptic function method [9], Weierstrass elliptic function method [7], Darboux and Backlund transform [21], symmetry reduction method [28], the tanh method [3], auxiliary method [16], extended tanh method [30], sine-cosine method [29], homogeneous balance method [10], exp-function method [34], the (G /G)expansion method [20], the modified simple equation method [17] first integral method [27], exp(−φ (ξ ) )−expansion method

34

A. Akbulut, F. Tas¸ can / Chaos, Solitons and Fractals 104 (2017) 33–40

[19], extended-tanh method [37], modified extended tanh-function method [33–36]. In this paper, we will search conservation laws with conservation theorem and then we will obtain exact traveling wave solutions with modified extended tanh-function method of (1+1)dimensional nonlinear coupled Klein–Gordon–Zakharov equation. In Section 2, we will define necessary preliminary informations about conservation theorem and modified extended tanh-function method. In Section 3, we will apply these methods to the (1+1)dimensional nonlinear coupled Klein–Gordon–Zakharov equation. In Section 3.1, firstly, we will reduce given system to different form. Then, we will obtain Lie symmetry generators, formal Lagrangian, adjoint equations of founded system. Next, we will find trivial conservation laws for this system. At last, we will convert founded conservation laws for the given system. In Section 3.2, we will reduce given system to nonlinear ordinary differential equation. Then, we will find exact traveling wave solutions in the form of hyperbolic, trigonometric, rational fınction solutions for (1+1)dimensional nonlinear coupled Klein–Gordon–Zakharov equation. Finally, some conclusions will be given.

Theorem 1. Every Lie point symmetry of Sys. (1) yields a conservation law for Sys. (1) and Sys. (8). Conserved vectors are described with following formulea

2. Preliminary informations

where u is an unknown function and P is a polynomial of u and its partial derivatives, in which the highest order derivatives and the nonlinear terms are involved derivatives. We will tell steps of modified extended tanh-function method. Step 1: We use the following wave transformation

2.1. Conservation laws Consider the kth order system of partial differential equations in the following form





E α x, u, u(1) , . . . , u(k ) = 0,

α = 1, . . . , m

(1)

here x = (x1 , x2 , . . . , xn ) are n independent variables, u = (u1 , u2 , . . . , um ) are m dependent variables and u(k) denotes kth order partial derivatives, i.e., uα = Di ( uα ), uα = D j Di ( uα ). Di ; i ij means differentiation with respect to xi , namely total differentiation operator. Total differentiation operator is shown that

Di =

∂ ∂ ∂ + uαi + uαi j α + . . . , i = 1, . . . , n. ∂ xi ∂ uα ∂uj

(2)

The infinitesimal generator for the governing Sys. (1) can be written as follow

X = ξi

∂ ∂ + ηα α , ∂ xi ∂u

(3)

where ξ i and ηα are called infinitesimal functions, i = 1, 2, . . . , n and α = 1, 2, . . . , m. The kth prolongation of the infinitesimal generator is

X (k )

=X +η

(1 )α i

∂

∂ uαi

+ ...+ η

(k )α i1 i2 ...ik

∂

∂ uαi1 i2 ...ik

,k ≥ 1

(4)

  ηi(1)α = Di ηα − Di ξ j uαj   )α ηi(1ki)2α...ik = Dik ηi(1ki−1 − Dik ξ j uαi1 i2 ...ik−1 j 2 ...ik−1

L = wα E α

(6)

δ uα here

=

∂ uα

+

(−1 ) Di1 . . . Dis

s≥1

s

∂

∂ uαi1 ...is

,

α = 1, . . . , m.

(7)

δ is variational derivative. Then, adjoint equation system δ uα

can be obtain with

(E α )∗ =

δL . δ uα

Theorem 2. The n−tuple vector T = (T 1 , T 2 , . . . , T n )is a conserved vector of Sys. (1) if Ti satisfies

 

Di T i = 0

(10)

[26–32]. 2.2. Modified extended tanh-function method Consider the following nonlinear partial differential equation

P (u, ut , ux , utt , uxx , uxt , . . . ) = 0

u(x, t ) = u(ξ ),

(11)

ξ = kx + wt

(12)

where k and w are constants. If we substitute wave transformations in the form of (12) in Eq. (11), then Eq. (11) reduces to following ordinary differential equation





F u, u , u , u , . . . = 0

(13)

where the prime denotes the derivation with respect to ξ . Step 2: Think that exact solution of Eq. (13) can be obtained in the following form

u ( ξ ) = a0 +

n  

−i

ai ( φ ( ξ ) ) + bi ( φ ( ξ ) ) i



(14)

i=1

here ai , bi (i = 0, 1, . . . , n ) are constansts that will be determined later such that an = 0 or bn = 0 and n is the balance term. We can determine positive integer balance term by balancing the highest order derivatives and the nonlinear terms. In Eq. (14), φ (ξ ) satisfies the Riccati equation

φ  ( ξ ) = b + ( φ ( ξ ) )2

(15)

where b is a constant. Eq. (15) admits following solutions Case 1: If b < 0, solution of Eq. (15) is

(16) Case 2: If b > 0, solution of Eq. (15) is obtained by

φ (ξ ) =



b tan

(8)

     bξ or φ (ξ ) = −b cot bξ .

(17)

Case 3: If b = 0, then

1

φ (ξ ) = − . ξ

where wα = wα (xi , uα ) new adjoint variables. The Euler–Lagrange operator is given by ∞ 

(9)

where i = 1, . . . , n, α = 1, . . . , m and W α = ηα − ξ j uαj . The conserved vectors are obtained from which Eq. (9) involves the arbitrary solutions wα of the adjoint equation and hence one obtains an infinite number of conservation laws for Sys. (1) by specifying wα .

(5)

here i, j = 1, 2, . . . , n and α = 1, 2, . . . , m and il = 1, 2, . . . , m for l = 1, 2, . . . , k and Di is the total differentiation operator. Sys. (1) has always formal Lagrangian by

∂

δL  δL + Di1 . . . Dis (W α ) α δ uαi δ uii ...is 1 s≥1

      φ (ξ ) = − −b tanh −bξ or φ (ξ ) = − −b coth −bξ .

where

δ

T i = ξ iL + W α

(18)

Step 3: If we substitute Eq. (14) and its derivatives in Eq. (13) with the help of Eq. (15), we can get a polynomial in φ i (ξ ) (i = 0, ±1, ±2, . . .). Setting all the coefficients of φ i (ξ ) equal to zero yields a set of over determined nonlinear algebraic equations for a0 , k, w ai , bi (i = 0, 1, . . . , n ). When we solve obtained nonlinear algebraic equations using the Maple software, we can construct a variety of exact solutions for the nonlinear partial differential Eq. (11).

A. Akbulut, F. Tas¸ can / Chaos, Solitons and Fractals 104 (2017) 33–40



       ∂L ∂L ∂L + Dx (W w ) + Dx W k ∂vxx ∂ wxx ∂ kxx     ∂L +Dx W v (26) ∂vxx

3. Applications



+W v −Dx

In this section, we will obtain conservation laws and exact traveling wave solutions of (1+1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation in the following form

utt − uxx + u + α uv = 0

    vtt − vxx − β | u |2 xx = 0

and

(19)

where α and β are arbitrary constants. The function u(x, t) is a complex function and the function v(x, t) is a real function [1]. Coupled Klein–Gordon–Zakharov equation is model systems to describe nonlinear interactions in plasma, where u(x, t) denotes the fast time scale component of electric field raised by electrons (more precisely it denotes its slowly varying envelope), v(x, t) denotes the ion density fluctuation [6].





T t = Lξ 2 + W w −Dt





3.1. Conservation laws of coupled Klein–Gordon–Zakharov equation In this subsection we will obtain conservation laws of Sys. (19) by using conservation theorem. We should reduce the function u(x, t) to real function. So, we can choose the function u(x, t) in the following form

u(x, t ) = w(x, t ) + ik(x, t ) u∗ (x, t ) = w(x, t ) − ik(x, t )

(20)

where w(x, t) and k(x, t) are real functions and u∗ (x, t) is a complex conjugate of u(x, t). If we substitute (20) in Sys. (19), we get following system

wtt − wxx + w + α wv = 0,

∂ ∂ ∂ ∂ 2 + 2αv ∂ +t −k −w − . ∂x ∂t ∂k ∂w α ∂v   vtt − vxx − β 2w2x + 2k2x + 2wwxx + 2kkxx



= −2β w2x + k2x + wwxx + kkxx

(23)

Divergence condition is satisfied with Theorem 2, conservation u + u∗ u − u∗ laws are trivial. If we substitute w = , k= in conser2 2i vation laws (28), we find trivial conservation laws of Sys. (19) in complex form as ∼

T1x =

β (ut∗ ux + ut u∗x + u∗ uxt + u∗xt u ) + vxt 1 ∗ (u ux − ut u∗x + uxt u∗ − u∗xt u ), 2i t

∼

T1t = −β (2u∗x ux + u∗xx u + u∗ uxx ) − vxx 1 (−u∗ uxx + u∗xx u ). 2i

Case 2: If we use Lie symmetry generator X2 =

(29)

∂ , we obtain ∂x

(30)

Divergence condition (10) is equal to zero, so conservation laws u + u∗ u − u∗ (30) are trivial. If we use w = , k= in (30), we can 2 2i obtain as complex form ∼

1 (utt u∗ − utt∗ u ) + vtt , 2i ∼ 1 T2t = (ux ut∗ − u∗x ut − uxt u∗ + uu∗xt ) − vxt . 2i

T2x =

Case 3: If we calculate conservation laws with X3 = w

(31)

∂ ∂ −k ∂k ∂w

for Sys. (21), we obtain zero conservation laws in both real and complex form

(24)

∼

T3x = 0 ⇒ T3x = 0 (25)

To find conservation laws of Sys. (21), we calculated formal Lagrangian and adjoint equation system for Sys. (21). Now, we will formulate for obtaining conservation laws of Sys. (21) with

T x = Lξ 1 + W w

(28)

T2t = −wxx kt + kx wt + wxt k − kxt w − vxt .

If we solve adjoint equation Sys. (24), we can get





T2x = −kwtt + wktt + vtt ,

F1∗ =

h = 1, m = −k, n = w.

(27)

∂ , we can find conservation laws with the ∂t

conservation laws in the following form

where m = m(x, t, w, k, v ), n = n(x, t, w, k, v ), and h = h(x, t, w, k, v ) are new adjoint variables. Adjoint equation system can be calculated with the help of formal Lagrangian (23) for Sys. (21) as

δL = m + α mv − 2β whxx + mtt − mxx = 0, δw δL F2∗ = = n + α nv − 2β khxx + ntt − nxx = 0, δk δL F3∗ = = α mw + α nk + htt − hxx = 0. δv



kwxx − wkxx − vxx .

(22)

L = m(wtt − wxx + w + α wv ) + n(ktt − kxx + k + α kv )



∂L ∂ ktt

+2β (wx wt + kx kt + kkxt + wwxt ), T1t

+

We can obtain formal Lagrangian of Sys. (21) with

+h



T1x = wt kx − kt wx − kwxt + wkxt + vxt

Sys. (21) admits the following four Lie symmetry generators

∂ ∂ ∂ ∂ X1 = , X = , X =w −k , ∂t 2 ∂ x 3 ∂k ∂w

 + W k −Dt

help of (26), (27) and (25) in the following form

+ (21)



  ∂L ∂L w + Dt (W ) ∂vtt ∂ wtt      k ∂ L  v ∂ L +Dt W + Dt W . ∂ ktt ∂vtt

ktt − kxx + k + α kv = 0,

  vtt − vxx − β 2w2x + 2k2x + 2wwxx + 2kkxx = 0.

∂L ∂ wtt



+W v −Dt

Case 1: When X1 =

X4 = x

35



 

 ∂L ∂L ∂L ∂L k − Dx +W − Dx ∂ wx ∂ wxx ∂ kx ∂ kxx

T3t

=0⇒

∼ T3t

(32)

= 0.

αv ∂ Case 4: If we use X4 = x ∂∂x + t ∂∂t − k ∂∂k − w ∂∂w − 2+2 α ∂v and with the same procedure above, we obtain conservation laws for reduced system in the following form

T4x = 2β t (wx wt + kx kt + wwxt + kkxt ) + x(vtt − kwtt + wktt ) +t (vxt + wt kx − kt wx − kwxt + wkxt ) + 3vx + 3wkx

36

A. Akbulut, F. Tas¸ can / Chaos, Solitons and Fractals 104 (2017) 33–40

−3kwx + 6β wwx + 6β kkx , T4t



= −2β t kkxx + wwxx +

w2x

k2x

+



here a0 , a1 , b1 are constants. When we substitute Eq. (41) with the help of Eq. (15) into Eq. (40), then the coefficients of φ (ξ )j equal to zero ( j = 0, ±1 ) , we obtain a system of algebraic equations

− 3vt

−x(vxt + kt wx − wt kx − kwxt + wkxt ) −t (vxx − kwxx + wkxx ) − 3kt w + 3wt k.

(33)

(T x )

(T t )

Conservation laws (33) are trivial because of Dx + Dt ∗ We can reduce conservation laws (33) with w = u+2u , k = follows in complex form ∼ T4x

=

βt (

= 0. u−u∗ as 2i



   αβ w2 b1 a21 + 2a20 a1 + a1 2a1 b1 + a20  2   2  2 2 2 2



αβ w b1 2a1 b1 +  2  2 2

+ k2 − w

k − w + 1 a1 + 2 a1 b k − w

2

+ k2 − w

+

ut∗ ux

∗

u∗xt u

uu∗x

+ k −w 2

(35)

here k, w ∈ C, U (x, t ) is a real valued function and w0 is an arbitrary constant, then the Sys. (19) is reduced to following system of ordinary differential equation



2a1 k2 − w2



v1 (x, t ) = 



 

V =



βw k2

−

w2

∓



= 0.

 U 2.

k2 − w2 + 1



k2

−

w2

 U+

αβ w2 k2 − w2

U ( ξ ) = a0 + a1 φ ( ξ ) +

( φ ( ξ ) )2

 2 2

3 2 U = 0

= 0, = 0.

a1 = 0,

b=−

k2 −w2 +1 , 2 (k2 −w2 )

b1 =



⎞





1

⎜ ⎟ − k2 − w2 + 1 ⎜ ⎟ αβ ⎜ ⎟  ×⎜±  ⎟ ⎝ ⎠ k2 − w2 + 1 k2 − w2 + 1   w tanh ξ 2 2 2 k2 − w2

⎛



1 



⎞2

⎜ ⎟ − k −w +1 ⎜ ⎟ αβ ⎟.  ⎜  ±   ⎜ ⎟ 2 2 k −w ⎝ ⎠ k2 − w2 + 1 k2 − w2 + 1   w tanh ξ 2 2 β w2

2

2

2 k2 − w2

When b > 0, solution of Sys. (19) is

u3 (x, t ) = exp {i(kx + wt + w0 )}

⎛



2



⎞



⎜ ⎟ − k2 − w2 + 1 ⎜ ⎟ αβ ⎜ ⎟  ×⎜∓   2  ⎟ 2 k − w2 + 1 ⎝ ⎠ k2 − w2 + 1   ξ − w tan 2 2 k −w

⎛

(40)

(41)

(42)

u2 (x, t ) = exp {i(kx + wt + w0 )}

k −w

(38)

where the prime denotes the derivation with respect to ξ . From Eq. (40) homogeneous balance is 1 and then the solution of Eq. (40) is in the following form

b1

 2

, k = k, w = w; w When b < 0, solution of Sys. (19) is

(39)



2

1 − 2αβ (k2 −w2 +1 )

When substituting Eq. (39) in first equation of Eq. (38), we obtain following ordinary differential equation

U  +

+ αβ w2 a31 = 0,

 2

 (a0 − b1 ξ )2



v2 (x, t ) = 

Integration second equation in (38) twice with respect to ξ and taking integration constant is zero for simplicity, we find 2

k2 − w2

(37)



k2 − w2 V  − β w2 U 2

β w2

where ξ = wx + kt + ξ0 . Case 2: a0 = 0,

k2 − w2 U  + k2 − w2 + 1 U + αUV = 0



2

+ αβ w2 b31 = 0,

k −w

in Sys. (36),(here ξ 0 is constant), it is obtained that



= 0,

u1 (x, t ) = exp {i(kx + wt + w0 )}(a0 − b1 ξ )

(36)

v(x, t ) = V (ξ ) = V (wx + kt + ξ0 )

= 0,

If we solve the Sys. (42), we find eight different solutions. Here we use only following five solutions because of a1 = 0 or b1 = 0 . Case 1: When a0 = a0 , a1 = 0, b = 0, b1 = b1 , k = ±i, w = 0, we obtain only following one solution because of b = 0.

Substituting wave transformations



2

3αβ w2 a0 b1 k2 − w

wUt − kUx = 0

U (x, t ) = U (ξ ) = U (wx + kt + ξ0 )

2

 2

3αβ w2 a0 a1 k2 − w2

⎛

Utt − Uxx + k2 − w2 + 1 U + αU v = 0 = 0.

+



k − w + 1 a0 = 0,

2



In this subsection we will construct exact traveling wave solutions of (1+1)-dimensional nonlinear coupled Klein–Gordon– Zakharov equation using modified extended tanh-function method. For applying this method, we should reduce Sys. (19) to system of ordinary differential equation. Hafez and Akbar reduced this equation with following transformation to system of ordinary differential equation [13]. If we consider transformation

xx



2b1 b k − w2

3.2. Exact solutions of coupled Klein–Gordon–Zakharov equation

vtt − vxx − β U



a1 b21

αβ w 4a0 a1 b1 + a0 2a1 b1 + a0  2  2  2 2

∼

 2

+

2a20 b1



T4t = −β t (2u∗x ux + u∗xx u + u∗ uxx ) − xvxt − t vxx x t (−u∗x ut + ut∗ ux + u∗xt u − u∗ uxt ) + (−u∗ uxx + u∗xx u ). (34) 2i 2i





 2

∗

u(x, t ) = exp {i(kx + wt + w0 )}U (x, t )

a20

k − w + 1 b1 + 2b1 b k2 − w2

+ uxt u + + u ux ) ) + 3β ( t ∗ ∗ +3vx + t vxt + xvtt − (ut ux − ut ux − uxt u∗ + u∗xt u ) 2i 3 ∗ x 3 ∗ ∗ (utt u − utt u ) − uux + u∗ ux , 2i 2i 2i ut u∗x

 

v3 (x, t ) = 

2 k2 − w2



2





⎞2

⎜ ⎟ − k −w +1 ⎜ ⎟ αβ ⎟ ⎜   ∓     ⎟ 2 2 k2 − w2 ⎜ 2 2 2 k −w +1 ⎝ ⎠ k −w +1  ξ − w tan 2 2 β w2

k −w

2

2

2 k2 − w2

here ξ = wx + kt + ξ0 . If we choose k2 − w2 + 1 = 0, b and b1 equal to zero, so we can’t obtain third solution because of a1 = b1 = 0.

A. Akbulut, F. Tas¸ can / Chaos, Solitons and Fractals 104 (2017) 33–40



Case 3: a0 = 0, a1 = ∓

2 k2 −w2 − αβ ( )

w

, b=−

When b < 0, first solution is

⎛ 

k2 −w2 +1 , 2 (k2 −w2 )



1

− ⎜ αβ ⎜ u4 (x, t ) = exp {i(kx + wt + w0 )} × ⎜ ± ⎜ ⎝

⎛ 

1



− ⎜ αβ ⎜ ⎜ v4 (x, t ) =  2 ± k − w2 ⎜ ⎝

β w2

37

b1 = 0, k = k, w = w;

k2 − w2 + 1 tanh k2 − w2



k2 − w2 + 1



2 k2 − w2



ξ



k −w 2

2

w

k2 − w2 + 1 tanh k2 − w2



k2 − w2 + 1



2 k2 − w2

ξ

 

k −w 2

2

w





⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎞2 ⎟ ⎟ ⎟. ⎟ ⎠

When b > 0, second solution is

⎛ 

 2

− ⎜ αβ ⎜ ⎜ u5 (x, t ) = exp {i(kx + wt + w0 )} × ⎜∓ ⎝

⎛ 

 2

− ⎜ αβ ⎜ ⎜ v5 (x, t ) =  2 ∓ k − w2 ⎜ ⎝

β w2



−



−

2 k2 − w2 + 1 k2 − w2

2 k2 − w2 + 1





k2 − w2

−

tan

k2 − w2 + 1



2 k2 − w2

ξ

 

2w



 tan

−

k2 − w2 + 1



2 k2 − w2

ξ





k −w 2

2w

2



k −w 2

2



⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎞2 ⎟ ⎟ ⎟ ⎟ ⎠

here ξ = wx + kt + ξ0 . In this case we can’t obtain third solution the same reason in case 2. Case 4: a0 = 0, a1 = ∓

2 k2 −w2 − αβ ( )

w

, b=−

w = w; When b < 0, we get

u6 (x, t ) = exp {i(kx + wt + w0 )}

⎛

⎛ 

⎜ ⎜ tanh ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜∓ ⎜ ⎜ ⎜ ⎜ ⎝

k2 −w2 +1 , 8 (k2 −w2 )

b1 = ±

2 k2 −w +1

2 4βα w − αβ

, k = k,

⎞ ⎛  2  ⎞2  ⎞2 2 k − w2 + 1 ξ⎟  ξ⎟ ⎟ ⎜  k2 − w2 k2 − w2 ⎟ k2 + 1 + k2 + 1 − tanh ⎜ ⎟ w2 − w2 ⎟ ⎟ ⎠ ⎝ ⎠ 4 4 ⎟ ⎟ ⎟  ⎟ ⎛  2  ⎞ 2 ⎟ 2 k −w +1  ⎟  ξ 2 2 ⎟ ⎜ ⎟ 2 2 1 k −w +1 k −w ⎟ ⎜ ⎟ 2 − βα w tanh 2 2 ⎠ ⎝ ⎠ αβ 4 k −w

2 k2 − w2 + 1

38

A. Akbulut, F. Tas¸ can / Chaos, Solitons and Fractals 104 (2017) 33–40

v6 (x, t ) = 

β w2 k2

− w2



⎛

⎛ 

⎞2 ⎛  2  ⎞2  ⎞2 2 k − w2 + 1 ξ⎟  ξ⎟ ⎟ ⎜  k2 − w2 k2 − w2 ⎟ k2 + 1 + k2 + 1 − tanh ⎜ ⎟ w2 − w2 ⎟ ⎟ ⎠ ⎝ ⎠ 4 4 ⎟ ⎟ ⎟ ⎟. ⎛  2 ⎞  2 ⎟ 2 k −w +1  ⎟  ξ 2 2 ⎟ ⎜ ⎟ 1 k −w +1 k2 − w2 ⎟ ⎜ ⎟ 2 − βα w tanh 2 2 ⎠ ⎝ ⎠ αβ 4 k −w

2 k2 − w2 + 1

⎜ ⎜ tanh ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜∓ ⎜ ⎜ ⎜ ⎜ ⎝

When b > 0, we get

u7 (x, t ) = exp {i(kx + wt + w0 )}

⎛

⎛

− ⎜ ⎜ tan ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜∓ ⎜ ⎜ ⎜ ⎜ ⎝

v7 (x, t ) = 

β w2 k2 − w2

⎛



⎞ ⎛  2  ⎞2  ⎞2 2 k − w2 + 1 ξ⎟  ξ⎟ ⎟ ⎜ −  k2 − w2 k2 − w2 ⎟ k2 + 1 − k2 − 1 − tan ⎜ ⎟ w2 + w2 ⎟ ⎟ ⎠ ⎝ ⎠ 4 4 ⎟ ⎟ ⎟ ⎟ ⎛  2 ⎞  2 ⎟ 2 k −w +1    ⎟  − ξ 2 2 ⎟ ⎜ ⎟ 2 k −w +1 2 k2 − w2 ⎟ ⎟  βα w tan ⎜ − −  ⎠ ⎝ ⎠ αβ 4 k2 − w2



⎞2 ⎛  2  ⎞2  ⎞2 2 k − w2 + 1 ξ⎟  ξ⎟ ⎟ ⎜ −  k2 − w2 k2 − w2 ⎟ k2 + 1 − k2 − 1 − tan ⎜ ⎟ w2 + w2 ⎟ ⎟ ⎠ ⎝ ⎠ 4 4 ⎟ ⎟ ⎟ ⎟. ⎛  2 ⎞  2 ⎟ 2 k −w +1    ⎟  − ξ 2 2 ⎟ ⎜ ⎟ 2 k −w +1 2 k2 − w2 ⎟ ⎜ ⎟   βα − − w tan ⎠ ⎝ ⎠ αβ 4 k2 − w2

2 k2 − w2 + 1

 ⎛

− ⎜ ⎜ tan ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜∓ ⎜ ⎜ ⎜ ⎜ ⎝



2 k2 − w2 + 1



When b = 0 k2 = w2 − 1 , we get

⎛  ⎜ ⎝

u8 (x, t ) = exp {i(kx + wt + w0 )}⎜∓

⎛  v8 (x, t ) = 

β w2 k2

−

w2

⎜ ⎜ ⎝∓

here ξ = wx + kt + ξ0 . Case 5: a0 = 0, a1 = ∓ w = w;

−

2

−

2

⎞

αβ ⎟ ⎟ wξ ⎠

⎞2

αβ ⎟ ⎟ wξ ⎠



2 k2 −w2 − αβ ( )

w

, b=

k2 −w2 +1 , 4 (k2 −w2 )

b1 = ±

2 k2 −w +1

2 2βα w − αβ

, k = k,

A. Akbulut, F. Tas¸ can / Chaos, Solitons and Fractals 104 (2017) 33–40

39

When b < 0,

u9 (x, t ) = exp {i(kx + wt + w0 )}

⎛

⎛

−

⎜ ⎜ tanh ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜± ⎜ ⎜ ⎜ ⎜ ⎝ v9 (x, t ) = 

β w2 k2 − w2

⎛

⎞2

k2 − w2 + 1 ξ⎟  ⎜  k2 − w2 ⎟ k2 + 1 − k2 − 1 − tanh ⎜ ⎠ ⎝ 2

 −

2



αβ

⎛ ⎜ k2 − w2 + 1 − βα w tanh ⎜ ⎝ k2 − w2

⎞

⎞2

−

k2 − w2 + 1 ξ⎟ ⎟ k2 − w2 ⎟ w2 + w2 ⎟ ⎟ ⎠ 2 ⎟

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞

−

k2 − w2 + 1 ξ⎟ k2 − w2 ⎟ ⎠ 2



⎛

⎛

⎜ ⎜ tanh ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜± ⎜ ⎜ ⎜ ⎜ ⎝

⎛

⎞2

k2 − w2 + 1 − ξ⎟  ⎜  k2 − w2 ⎟ k2 + 1 − k2 − 1 − tanh ⎜ ⎠ ⎝ 2

 −

2

αβ



⎛ ⎜ k2 − w2 + 1 − βα w tanh ⎜ ⎝ k2 − w2

⎞2

⎞2

k2 − w2 + 1 − ξ⎟ ⎟ k2 − w2 ⎟ w2 + w2 ⎟ ⎟ ⎠ 2 ⎟

⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠

⎞

−

k2 − w2 + 1 ξ⎟ k2 − w2 ⎟ ⎠ 2

When b > 0,

u10 (x, t ) = exp {i(kx + wt + w0 )}

⎛

⎛

⎜ ⎜ tan ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜∓ ⎜ ⎜ ⎜ ⎜ ⎝ v10 (x, t ) = 

β w2 k2 − w2

⎛

⎛

⎞2

k2 − w2 + 1 ξ⎟  ⎜  k2 − w2 ⎟ k2 + 1 + k2 + 1 − tan ⎜ ⎠ ⎝ 2

 −

2



αβ

⎛ ⎜ k −w +1 βα w tan ⎜ ⎝ k2 − w2 2

2

⎞2

⎞

k2 − w2 + 1 ξ⎟ ⎟ k2 − w2 ⎟ w2 − w2 ⎟ ⎟ ⎠ 2 ⎟

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎞

k2 − w2 + 1 ξ⎟ k2 − w2 ⎟ ⎠ 2

 ⎛

⎜ ⎜ tan ⎜ ⎜ ⎜ ⎝ ⎜ ⎜ ⎜ ×⎜∓ ⎜ ⎜ ⎜ ⎜ ⎝

⎛

⎞2

k2 − w2 + 1 ξ⎟  ⎜  k2 − w2 ⎟ k2 + 1 + k2 + 1 − tan ⎜ ⎠ ⎝ 2

 −

2

αβ



⎛ ⎜ k2 − w2 + 1 βα w tan ⎜ ⎝ k2 − w2

⎞2

⎞2

k2 − w2 + 1 ξ⎟ ⎟ k2 − w2 ⎟ w2 − w2 ⎟ ⎟ ⎠ 2 ⎟

⎞

k2 − w2 + 1 ξ⎟ k2 − w2 ⎟ ⎠ 2

⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠

40

A. Akbulut, F. Tas¸ can / Chaos, Solitons and Fractals 104 (2017) 33–40





When b = 0 k2 = w2 − 1 ,

u11 (x, t ) = exp {i(kx + wt + w0 )}

⎛ ⎜ ⎝

×⎜∓



4wξ βα

⎛ v11 (x, t ) = 

β w2 k2

−

⎞

   √  2 −4 + ξ 2 w2 − 1 − ξ 2 w2 + ξ 2 ⎟

w2

⎜ ⎜ ⎝∓

−

⎟ ⎠

1

αβ

⎞2

   √  2 −4 + ξ 2 w2 − 1 − ξ 2 w2 + ξ 2 ⎟ 

4wξ βα

−

1

⎟ ⎠

αβ

here ξ = wx + kt + ξ0 . 4. Conclusion In this paper, we dealt with (1+1)-dimensional nonlinear coupled Klein–Gordon–Zakharov equation. In Section 3.1, we used conservation theorem for obtaining conservation laws of coupled Klein–Gordon–Zakharov equation. Our system inherits complex funtion u(x, t), so we reduced this system to different system which have only real functions. Then for reduced system, we obtained Lie symmetry generators, formal Lagrangian and adjoint system. We found solutions of adjoint system because of obtaining local conservation laws. Then we obtained trivial conservation laws for reduced system. We have written in the form these conservation laws the functions u(x, t), u∗ (x, t) and v(x, t). As a result of this study we saw that we can find conservation laws containing complex functions. In Section 3.2, we utilized modified extended tanh-function method to coupled Klein–Gordon–Zakharov equation for finding traveling wave solution of this system. Firstly we reduced complex system to nonlinear ordinary differential equation. Then we applied method to founded ordinary differential equation. Consequently, we found eleven exact traveling wave solutions which is different from each other. These obtained solutions contain hyperbolic function solutions, trigonometric function solutions, rational solutions. Obtained solutions are useful for applications in mathematical physics and engineering. Thus, we conclude that the proposed methods can be extended to solve the different nonlinear complex partial differential equations, nonlinear partial differential equations and nonlinear fractional order differential equations. References [1] Alquran M, Katatbeh Q, Al-Shrida B. Applications of first integral method to some complex nonlinear evolution systems. Appl Math Inf Sci 2015;9:825–31. [2] Anco SC, Bluman GW. Direct construction method for conservation laws of partial differential equations. part II: general treatment. Eur J Appl Math 2002;9:567–85. [3] Bekir A, Cevikel AC. Solitary wave solutions of two nonlinear physical models by tanh–coth method. Com Non Sci Numer Simul 2009;14:1804–9. [4] Biswas A, Kara AH, Bokhari AH, Zaman FD. Solitons and conservation laws of Klein–Gordon equation with power law and log law nonlinearities. Nonlinear Dyn 2013;73:2191–6. [5] Buhem E, Bluman G, Kara AH. Conservation laws for some systems of nonlinear PDEs via the symmetry/adjoint symmetry pair method. J Math Anal Appl 2016;436:94–103. [6] Chen J., Liu L., Liu L. Separation transformation and a class of exact solutions to the higher-dimensional Klein-Gordon-Zakharov equation. Adv Math Phys 2014, Article ID 974050, 8 pages. [7] Chen Y, Yan Z. The weierstrass elliptic function expansion method and its applications in nonlinear wave equations. Chaos Solitons Fractals 2006;29:948–64.

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