Binary simplest equation method to the generalized Sinh–Gordon equation

Accepted Manuscript Title: Dispersive dark optical soliton with the generalized sinh–Gordon equation Author: A. Neirameh PII: DOI: Reference:

S0030-4026(15)00680-4 http://dx.doi.org/doi:10.1016/j.ijleo.2015.07.128 IJLEO 55843

To appear in: Received date: Accepted date:

26-6-2014 22-7-2015

Please cite this article as: A. Neirameh, Dispersive dark optical soliton with the generalized sinhndashGordon equation, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.07.128 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Dispersive dark optical soliton with the generalized sinh–Gordon equation

A. Neirameh, Department of Mathematics, Faculty of sciences, Gonbad Kavoos University, Gonbad, Iran

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Abstract:

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In this work dispersive optical solitons that are governed by the generalized sinh–Gordon equation. We successfully construct the new exact double periodic traveling wave solutions of the generalized Sinh– Gordon equation by introducing the binary simplest equation method. The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic solutions of the generalized Sinh–Gordon equation is solitary wave solutions and simpler than other methods.

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Key Words: double periodic exact wave solutions, binary simplest equation method; generalized sinh– Gordon equation.

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1. Introduction :

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Phenomena in physics and other fields are often described by nonlinear evolution equations. When we want to understand the physical mechanism of phenomena in nature, described by nonlinear evolution equations, exact solutions for the nonlinear evolution equations have to be explored. Nonlinear evolution equations are special classes of crucial mathematical model to describe many nonlinear physical phenomena. Traveling wave solutions of these equations play an important role in nonlinear science, for these solutions may well reflect inner aspects of nonlinear problems and can be easily used in further applications. One of the great interests is the problem of finding exact soliton solutions of the integrable nonlinear models. Based on these exact solutions directly, we can accurately analyze the properties of propagating soliton pulses in nonlinear physical systems. This area of study has been put to good use for the past few decades. In the recent years, seeking exact solutions of nonlinear partial differential equations (NLPDEs) is of great significance, since the nonlinear complex physical phenomena related to the NLPDEs are involved in many fields from physics, mechanics, biology, chemistry and engineering. To this aim, a vast variety of powerful and direct methods for finding the exact significant solutions of NLPDEs though it is rather difficult have been derived. Some of the most important methods are tanhextended tanh method [6-8], solitary wave ansatze method [9–11], tanh method [12,13], multiple expfunction method [14], Kudryashov method [15–16], Hirota’s direct method [17,18], transformed rational function method [19] and others [20]. They produce many kinds of exact solutions to a given evolution equations. However, their application and implementation are not direct and easy. Conversely, the tanh-function method, the homogeneous balance method, the Jacobi elliptic function expansion method, the sub-ODE method, the F-expansion method, the Sine–Cosine function method, the Exp-function method, the projective Riccati equation method, the simplest equation method etc. are modern algebraically methods. They are directly used to produce many kinds of exact solutions for a given NLEEs and systematically implemented by computer algebra system. Moreover these methods are favored by applied researchers. The layout of this paper is as follows: in Sect. 2, we briefly introduce the binary simplest equation method. In Section 3, we construct a series of abundant double periodic exact travelling wave solutions of generalized sinh–Gordon equation. A summary is given in the final section 4.

Page 1 of 10

2. The binary simplest equation method In this section we introduce the binary simplest equation method. For this aim at first we consider the nonlinear partial differential equation (f (u ),u x ,u t ,u xx ,u tt ,u xt ,...)  0, (1)

ip t

Where f u  is a composite function which is similar to sin(nu ) or sinh(nu ), (n  1, 2,...) etc. The binary simplest equation method is simply represented as follows: We make a transformation  U ( )  u    ,   1 (x  c1t),   2 (x  c2 t), V ( )  Based on this we use the following changes

2 2 .  c 2  . 2   x  2

an

2 2 .  c1 2 . 2   x 

cr

,

  .  c 2 . , t    .  c 2 . , x 

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  .  c1 . , t    .  c1 . , x 

(2)

Where 1 , 2 ,c1 ,c2 are unknown parameters which to be determined later. Substituting (2) into (1), yields

 (U ,U ,U ,...,V ,V ,V ,...)  0.

M

(3)

Then, with the aid of simplest equation method defined in follow, we can obtained the solutions of Eq. (3) at with substituting into (2), solutions of Eq.(1) can be obtained.

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Eq. (3) in a finite series

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2.1 Simplest equation method Step1. The basic idea of the simplest equation method consists in expanding the solutions U and V of l

m

al  0, and V ( )   s j y j ,

U ( )   ai z i , i 0

j 0

s m  0,

(4)

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where the coefficients ai , am are independent of  , and z  z ( ), y  y   are the functions that satisfy some ordinary differential equations. In this paper, we use the Bernoulli equation [20] as simplest equation dz  2   ,     bz  az d dy  ay    by 2   , d

Eq. (5) admits the following exact solutions z    y   

a exp a     0   , 1  b exp a    0   a exp a   0  1  b exp a   0  

(5)

(6) ,

for the case a  0, b  0, a  0, b  0 and z    y   

a exp a    0  , 1  b exp a    0  a exp a   0   1  b exp a   0  

(7) ,

Page 2 of 10

for the case a  0, b  0, a  0, b  0 , where 0 ,0 is a constant of integration. Remark1. l and m are positive integer, in most cases, that will be determined. To determine the parameters l and m ; we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. Step2. Substituting (4) into (3) with (5), then the left hand side of Eq. (3) is converted into a polynomial in z   , y   ; equating each coefficient of the polynomial to zero yields a set of algebraic equations

Eq. (8) admits the following exact solutions

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dz  Az    z 2   , d dy  Az    z 2   , d

ip t

for ai ,si , a, a, b, a , b , c . Step3. Solving the algebraic equations obtained in step 2, and substituting the results into (4), then we obtain the exact traveling wave solutions for Eq. (1). Remark2. In Eq. (5), when a  A , a  A and b  1, b  1 we obtain the Bernoulli equation

 A  1  tanh  2   0    ,   

A y    2

 A  1  tanh    0    , 2  

A 2

 A  1  tanh  2    0    ,   

an

A 2

z   

z   

M

when A  0, A  0 , and

(10)

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when A  0, A  0 .

(9)

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A  A y    1  tanh    0    , 2 2 

(8)

Remark3. This method is a simple case of the Ma- Fuchssteiner method [4].

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3. Our idea to the generalized sinh–Gordon equation

In this section, we further consider the following generalized Sinh–Gordon equation: u tt  au xx  b sinh(nu )  0,

(11)

Wazwaz [5] studied this equation. Where a , b are two constants and n is a positive integer. First, considering the following transformation: a    (x  ct),    (x  t), a  c 2 , (12) c where ,c are two parameters to be determined later, under the transformation (11), Eq. (1) can be rewritten as  2c 2 (c 2  a ) u    2a (a  c 2 ) u  bc 2 sinh(nu)  0. (13) By means of a similar ansatz as given in Refs. [1-3], letting u

4 U ( ) tanh 1 ( ), n V ( )

(14)

for Eq. (13), yields U U      2c 2 (c 2  a )  (U 2 V 2 )   2U 2    2a (a  c 2 )  (V 2  U 2 )   2U 2  u  nbc 2 (U 2 V 2 ). U U    

Page 3 of 10

Successive differentiations of this result with respect to both  and  results in

c 2 U  a V  ( )  ( )  0, UU  U VV V

(15)

and from (15), one has

ip t

c 2 U  a V  ( )  ( )   , UU  U VV V where  is a parameter to be determined later, i.e.

 4 (16) U  1U 2  1 , 4c 2  (17) V 2  V 4   2V 2   2 , 4a where 1 ,1 , 2 , 2 , are integral constants. Considering (14-17) we have the corresponding constraint conditions

us

cr

U 2 

c 2 1  a 2  0

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 2c 2 (c 2  a ) 1   2a (a  c 2 ) 2  nbc 2  0,

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Next, we study Eqs. (16) and (17) under the conditions of Simplest equation method. By making balance 2 2 between the linear terms U 4 ,V 4 and the nonlinear terms U  , V   respectively to determine the value of l , m , we have got that l  1 and m  1 , so that U ( )  a1 z     a0 , (18) V ( )  s1 y    s 0 ,

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d

Substituting (18) along with (5) in Eq. (16)-(17) and then setting the coefficients of z i (i  0,1, 2,3, 4) and y j (j  0,1, 2,3, 4) to zero in the resultant expression, we obtain two set of algebraic equations and by solving these equations with the aid of Maple we have

Case1: In this case we consider

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a0  



1 b   2





,

a1 2  1c  12c 2   1

 2 c  3 1

a  



2  1c  12c 2   1 c





 c   1 c 2 2

1

c 2 1c  3 12c 2   1





s0   1 , b  2



2  2 a   22c 2   2





s1 2  2 a   22a   2

a ,



 2 a  3  a    a a  2  a  3  a    2 2

2

a 



2

(19)

2

2 2

2

c a So from (14), (18) and (19) along with (6) we have solutions of (11) as follows

Page 4 of 10

4 tanh 1 n

u1 









   2 2 c 2 1c  3  12c 2   1   c 2 1c  3  1 c   1   exp    (x  ct)  0    c c   2  1c   12c 2   1 c     a1      2 2 a 2  1c   12c 2   1  c 2 1c  3  1 c   1   1 1 1 exp     (x  ct)  0   2 c    21c  3  12c 2   1 c        a 2 2 a  3  22a   2 a 2 2 a  3 22a   2   a     exp      (x  t)  0  c a a     2   2 a   22c 2   2      s  1     s 2   2 a   22a   2 a 2 2 a  3 22a   2   a    1 1 1 exp     (x  t)  0    c 2 a    2 2 a  3 22a   2 a     Where s1 and a1 are arbitrary.



















a1 2 1c  12c 2   1





s0  



1 , b  2



d

 2 1c  3  1 c   1 c 2 2

Ac ce p

c



te

c 2 1c  3  12c 2   1

a  

,

M





2 1c   12c 2   1 c

1 b   2





Case2: For this case we have a0  

ip t







us







an







cr





a 



2 2 a   22c 2   2





a ,



s1 2  2 a   22a   2



 2 2 a  3  a   2 2 2





 a 2 2 a  3  22a   2

           .    a        



(20)

a



a

And from (14), (18) and (20) along with (6) we have solutions of (11) as follows u2 

4 tanh 1 n









   2 2 c 2 1c  3  12c 2   1   c 2 1c  3 1 c   1   exp    (x  ct)  0   c c   2 1c  12c 2   1 c      a1     2 2 a 2  1c   12c 2   1  c 2 1c  3 1 c   1   1 1 1 exp     (x  ct)  0  c 2     2 1c  3 12c 2   1 c         a 2  2 a  3  22a   2  a 2 2 a  3  22a   2   a      exp     (x  t)  0  c a a     2 2 a   22c 2   2      s  1     s 2  2 a   22a   2  a 2 2 a  3  22a   2  a    1 1 1 exp     (x  t)  0    2 c a     2  2 a  3 22a   2 a     Where s1 and a1 are arbitrary.









































           .    a        

Page 5 of 10

Case3: We consider the following values 2 1 ,a  2 1 ,b   a0  c 

1 1     , s   2a 2 , a  2 ,b   0 2  2c 2 a

 

(21)

ip t

From (14), (18) and (21) along with (6) we have solutions of (11) as follows

an

us

cr

        2  exp 2   (x ct)  1 1 0  2 1    c  a1   1          1 exp  2 1   (x  ct)   0   4  1  2 c u 3  tanh  . n  a     2  2 exp  2  2   (x  t)  0   c 2a  2        s1   1      exp  2   (x  a t)      1  2  0     c 2 a     

2c  21 , a0  c , A  21  

,

s1 

2 a 2 2 ,s0  c , A  2  2  

(22)

d

a1 

M

Where s1 and a1 are arbitrary. In the same way, using solution (9) of Eq. (8), ansatz (18), we obtain the following exact solutions of Eq. (11) as follows: Case4: In this case from sets of algebraic equations as above we obtain

So from (14), (18) and (22) along with (9) we have solutions of (11) as follows

Ac ce p

te

 c    2 1  2 1    2 1 1  tanh   (x  c t)   0     c   2       4    u 4  tanh 1  , n   2  2   a 2  2  a   2 2 1  tanh   (x  t)  0     c    2     c        

Case5: For this case we consider

a0   a1 





2c  

A 



2  1c   12c 2   1 c



c 2 1c  3 12c 2   1 c

,

s0  

, s1 





2  2 a   22c 2   2

a ,



2 a 

A 



(23)



a 2 2 a  3  22a   2



a

From (14), (18) and (23) along with (9) we have solutions of (11) as follows

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



 2  1c  12c 2   1 c    2 1   2 2   c 2 1c  3 1 c   1 1  tanh    (x  c t)  0          4  2   1 u 5  tanh  n  2  2 a   22c 2   2   2  2  a     2 a 2 2 a  3 2 a   2 1  tanh    (x  t)  0         c      2  





2c  

A 

s0  

, s1 



c 2 1c  3 12c 2   1





2 a , 

A 

c

 



a

,

cr



,

2 2 a   22c 2   2

(24)

.

us

a1 



 a 2 2 a  3 22a   2 a

an

a0  



ip t

Case6: Also we have 2 1c  12c 2   1 c









From (14), (18) and (24) along with (9) we have solutions of (11) as follows





 2 1c  12c 2   1 c    2 1   2 2   c 2 1c  3 1 c   1 1  tanh    (x  c t)  0      2     4    1 u 6  tanh  n  2 2 a   22c 2   2   2  2  a     2    a 2  2 a  3 2 a   2 1  tanh  2   (x  c t)  0            



M









    , a    

d



    , a    

Figures caption: Double periodic soliton solutions related to case 6 are shown in Figs. (a–d).

Ac ce p

region x  4...4, t  0...8 ,

te

Figure a: Solitary wave variation of u 6 for c  1, a  1, 1  1, 1  2,   4,  2  1, 2  2 in

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Fig.a Figure b: Solitary wave variation of

u 6 for c  1, a  2, 1  2, 1  2,   4,  2  2, 2  2 in

M

an

us

cr

ip t

region x  4...4, t  0...8 ,

te

Ac ce p

region x  4...4, t  0...8 ,

d

Fig.b Figure c: Solitary wave variation of u 6 for c  1, a  3, 1  3, 1  3,   4,  2  1, 2  3 in

Fig.c

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Figure d: Solitary wave variation of

u 6 for c  1, a  4, 1  6, 1  4,   4,  2  6, 2  4 in

M

an

us

cr

ip t

region x  4...4, t  0...8 ,

4. Conclusions:

d

Fig.d

Ac ce p

te

In summary, we have developed successfully introduce the binary simplest equation method and obtained wider classes of exact traveling wave solutions for the generalized sinh–Gordon equation by using this binary method. Compared with the solutions obtained by using other methods, we obtained new exact solutions for equation (11).This implies that our method is more powerful and effective in finding the exact solutions of NLEEs in mathematical physics. We hope this method can be more effectively used to solve many nonlinear partial differential equations in applied mathematics, engineering and mathematical physics.

References:

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Ac ce p

te

d

M

an

us

cr

ip t

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