Topological soliton solutions to the coupled Schrodinger–Boussinesq equation by the SEM

Accepted Manuscript Title: Topological soliton solutions to the coupled Schrodinger–Boussinesq equation by the SEM Author: A. Neirameh PII: DOI: Reference:

S0030-4026(15)00787-1 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.028 IJLEO 55950

To appear in: Received date: Accepted date:

20-7-2014 4-8-2015

Please cite this article as: A. Neirameh, Topological soliton solutions to the coupled SchrodingerndashBoussinesq equation by the SEM, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.028 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.

Topological soliton solutions to the coupled Schrodinger–Boussinesq equation by the SEM A. Neirameh

ip t

Department of Mathematics, Faculty of sciences, University of Gonbad e Kavoos, Gonbad, Iran

cr

Abstract:

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In this present study we consider the coupled Schrodinger–Boussinesq equation [1]

iE t  E xx  1E  NE , 2  3N tt  N xxxx  3  N xx   2 N xx  E

  2

,

(1)

xx

an

Where 1 and 2 are real constants, and E  x , t  is a complex function, N  x , t  is a real

1. Introduction:

d

M

function. In this paper, we enhanced new traveling wave solutions of coupled Schrodinger– Boussinesq equation via simplest equation method. Implementation of the method for searching exact solutions of the equation provided many new solutions which can be used to employ some practically physical and mechanical phenomena. Keywords: coupled Schrodinger–Boussinesq equation; simplest equation method; soliton solution.

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The Schrodinger equation was proposed to model a system when the quantum effect was considered. For a system with N atoms, the Schrodinger equation is defined in 3N+ 1 dimension. With such high dimensions, even use today’s super computer, it is impossible to solve the Schrodinger equation for dynamics of N atoms with N >10. After assumed Hatree-Fork ansatz, the 3N+1 dimensions linear Schrodinger equation was approximated by a 3+1 dimensions nonlinear Schrodinger equation (NLSE) or Schrodinger-Poisson (S-P) system. Although nonlinearity in NLSE brought some new difficulties, but the dimensions were reduced significantly compared with the original problem. This opened a light to study dynamics of N atoms when N is large. Later, it was found that NLSE had applications in different subjects, e.g. quantum mechanics, solid state physics, condensed matter physics, quantum chemistry, nonlinear optics, wave propagation, optical communication, protein folding and bending, semiconductor industry, laser propagation, nano technology and industry, biology etc. Currently, the study of NLSE including analysis, numerics and applications becomes a very important subject in applied and computational mathematics. This study has very important impact to the progress of other science and technology subjects. Therefore, investigation of exact traveling wave solutions is becoming successively attractive in nonlinear sciences day by day. However, not all equations posed of these models are solvable. Hence it becomes increasingly important to be familiar with all traditional and recently developed methods for solving these models and the implementation of new methods. As a result, many new techniques have been successfully developed by diverse groups of

Page 1 of 9

mathematicians and physicists, such as, the Kudryashov method [2], the Exp-function method [3–6], the (G’/G)-expansion method [7–11] and some useful studies in this field[12-18]. The objective of this article is to apply the simplest equation method to construct the exact solutions for nonlinear evolution equations in mathematical physics via the KP equation.

ip t

2. The simplest equation method Step1. We first consider a general form of nonlinear equation

E u , u t ,u x ,u tt ,...  0.

(2)

cr

Step2. To find the traveling wave solution of Eq. (2) we introduce the wave variable   x  ct , so that

u  x ,t   u   ,

us

Based on this we use the following changes

an

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

(3)

(4)

and so on for other derivatives. Using (4) changes the PDE (2) to an ODE

M

2 2   c .    . x 2  2

Ac ce p

(5) in a finite series

te

d

 y  2 y  (5)   y , , 2 ,...   0,     where y  y ( ) is an unknown function,  is a polynomial in the variable y and its derivatives. Step3.The basic idea of the simplest equation method consists in expanding the solutions y ( ) of Eq. l

y ( )   ai z i , i 0

al  0,

(6)

where the coefficients ai are independent of  and z  z ( ) are the functions that satisfy some ordinary differential equations. In this paper, we use the Bernoulli equation [19] as simplest equation

dz  az    bz 2   , d

Eq. (7) admits the following exact solutions

z   

for the case a  0, b  0 and

z   

(7)

a exp a    0  

,

(8)

a exp a   0  

,

(9)

1  b exp a   0  

1  b exp a   0  

for the case a  0, b  0 , where 0 is a constant of integration. Remark1. l is a positive integer, in most cases, that will be determined. To determine the parameter l ; we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms.

Page 2 of 9

Step4.Substituting (6) into (5) with (7), then the left hand side of Eq. (5) is converted into a polynomial in z   ; equating each coefficient of the polynomial to zero yields a set of algebraic equations for ai , a , b , c . Step5. Solving the algebraic equations obtained in step 4, and substituting the results into (6), then we obtain the exact traveling wave solutions for Eq. (2).

ip t

Remark2. In Eq. (7), when a  A and b  1 we obtain the Bernoulli equation

dz  Az    z 2   , d

(10)

A 2

when A  0 , and

z   

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

A 2

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

(11)

(12)

an

when A  0 .

us

z   

cr

Eq. (10) admits the following exact solutions

M

Remark3.This method is a simple case of the Ma- Fuchssteiner method [20].

3. SEM to the coupled Schrodinger–Boussinesq equation

te

d

Eq. (1) are known to describe various physical processes in Laser and plasma, such as formation, Langmuir field amplitude and intense electromagnetic waves and modulational instabilities [11–14].In the following, as applications of the proposed method we construct many new exact solutions of Eq. (1) by using the method. In order to look for the exact traveling wave solutions of Eq. (1), we suppose that

Ac ce p

E  x , t   u  x , t  e i  ,  kx  ct  c 0 ,

(13)

Where u  x , t  is a real function k , c , c 0 are real constants. By substituting (13) into (1), we obtain

u t  2ku x  0,

(14)

u xx  c  k 2  1  u  Nu ,

3N tt  N xxxx  3  N

xx 2

(15)

  2 N xx  u 2  . xx

(16)

According to Eq. (14), we suppose that

u  x , t   u     u  x  2kt  c1  ,

(17)

Where c1 is the arbitrary constant. By substituting (17) into (15), we obtain

u     c  k 2  1  u  

(18)

N  x , t   v    v  x  2kt  c1  .

(19)

N  x ,t   We can suppose that

Page 3 of 9

By substituting (17) and (19) into (15) and (16), we obtain ODEs

u     c  k 2  1 u    u  v    0,

(20)

v     12 k 2   2 v    3v 2    u 2    C  0,

(21)

ip t

Where C is the integrationconstant to be determined later.By considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eqs. (20) and (21), we obtain m  2 for u and n  2 for v.Suppose that the solutions for Eqs. (20) and (21) can be expressed in the following form

u  a2 z 2  a1z  a0 ,

(22)

cr

v  b 2 z 2  b1z  b 0 ,

(23)

By substituting (22) and (23) into Eqs. (20) and (21) using the second order linear ODE

us

expressions (6) and (7), collecting all terms with the same order of z j , j  0,1, 2 . Theleft-hand side of Eqs. (20) and (21) is converted into another polynomial in z j . Equating each coefficient of this different power terms to zero yields a set of nonlinear algebraic equations for

an

a j , b j  j  0,1, 2  , k , c , 1 and 2 .With the aid of Maple, we obtain four classes of solutions of

this nonlinear algebraic system, namely,

17k 2  5c  51   2 10b 18k 2  6c  6 1   2

, a2 

3b 17 k 2  5c  51   2 

20 18k 2  6c  6 1   2  18k 2  6c  6 1   2

,

d

a1  0, a0  

M

Case1:

Ac ce p

a   18k 2  6c  6 1   2

te

b 2  6b 2 , b1  10 18k 2  6c  6 1   2 b , b0  1  c  k 2 ,

from (22) and (23) along with (8) we obtain u 1,1 

3b 17 k 2  5c  51   2 

20 18k 2  6c  6 1   2  18k 2  6c  6 1   2

  18k 2  6c  6 1   2  exp  2a    0     1  b exp   18k 2  6c  6 1   2    0      2 17 k  5c  51   2 , 10b 18k 2  6c  6 1   2







    2  

Page 4 of 9

v 1,1  6b

2

18k

2

 6c  6 1   2  exp  2 18k 2  6c  6 1   2    0    

1  b exp 18k

2

 6c  6 1   2     0  



2

18k 2  6c  6 1   2 exp   18k 2  6c  6 1   2    0     10 18k  6c  6 1   2 b 2   1  b exp  18k  6c  6 1   2   0    2 1  c  k ,

ip t

2

20 18k 2  6c  61   2  18k 2  6c  61   2



us

E 1,1  x , t  

3b 17k 2  5c  51   2 

cr

So from (13)we have

 2 2    18k  6c  61   2  exp  2 18k  6c  61   2  kx  ct  c 0  0    i  kx ct c0    e 2 2     1  b exp  18k  6c  61   2  kx  ct  c 0  0       2 17k  5c  51   2 e i  kx ct c0  , 2 10b 18k  6c  61   2 And from (17) we obtain 2

 6c  61   2  exp  2 18k 2  6c  61   2  x  2kt  c1  0    

d

18k

1  b exp 18k

te

N 1,1  x , t   6b

2



M

an



2

 6c  61   2   x  2kt  c1  0  



2

Ac ce p

18k 2  6c  61   2 exp   18k 2  6c  61   2  x  2kt  c1  0     10 18k  6c  61   2 b 2   1  b exp  18k  6c  61   2  x  2kt  c1  0    2 1  c  k , 2

Case2:

2 2 2 1 a0  0, a1  b  k 2  c  1   2 , a2   3 3 3 6

5ab 2 , 2 2 2 2 1 6  k  c  1   2 3 3 3 6

1 b 0  0, b1  0, b 2  b 2 , a   c  k 2  1 , 6 In this case solitary wave solutions for Eq. (1) are

Page 5 of 9

E 2,1  x , t   

2



2 2 2 1 6  k 2  c  1   2 3 3 3 6



 1  exp 2   c  k 2  1  kx  ct  c 0  0     e i  kx ct c 0  

1  b exp   c  k 2  1  kx  ct  c 0   0    

c  k 2  1 exp   c  k 2  1  kx  ct  c 0  0     e i  kx ct c 0  , 1  b exp   c  k 2  1  kx  ct  c 0   0    

cr

2 2 2 1 b  k 2  c  1   2 3 3 3 6



2

ip t

c  k

5 c  k 2  1b 2

us

and

2 2   1 2 c  k  1  exp  2 c  k  1  x  2kt  c1   0   , N 2,1 (x, t)  b 2 6 2   1  b exp  c  k  1  x  2kt  c1   0   

an



M

Case3:



a0  30k 2 1  30k 2c  63k 4   2 1   2c  3 2 k 2  312  6 1c  3c 2  C , a1  0, a2  0, b 2  0, b1  3 18k 2  6c  6 1   2 b , b 0  1  c  3k 2 ,

te

From (8) and (22) we obtain

d

a   18k 2  6c  6 1   2

and

Ac ce p

E 3,1  x , t   30k 2 1  30k 2c  63k 4  2 1  2c  32 k 2  312  61c  3c 2  C e i  kx ct c0  ,

N 3,1 (x, t)  3 18k 2  6c  6 1   2 b  18k 2  6c  6 1   2 b exp   18k 2  6c  6 1   2 b  x  2kt  c1  0    

1  b exp  18k

2

 6c  6 1   2  x  2kt  c1   0   



2

1  c  3k 2 ,

As above we obtain N 1,2 , N 2,2 , N 3,2 , E 1,2 , E 2,2 and E 3,2 from (7).

Case 4: According to the remark 2 by substituting (7),(22) and (23) along with (10) in (20) and (21)we obtain following coefficients

Page 6 of 9

a0  8 18k 2   2  6 1  6c  , a1  0, a2  12, b 0  1  c  k 2 , b1 

30 18k 2   2  6 1  6c , b 2  6. 19

A  18k 2   2  6 1  6c .

E 4,1  x , t   3 18k 2   2  6 1  6c   2

ip t

So from (11), (13) and (22) we obtain

,

us

i  kx ct c 0 

an

8 18k 2   2  6 1  6c  e

cr

  18k 2    6   6c   i kx ct c 2 1 0 1  tanh  kx  ct  c 0  0    e      2     

 18k 2    6   6c  3 18k 2   2  6 1  6c   2 1 1  tanh  N 4,1 (x, t)  x  2kt  c1   0        2 2     2

d

M

 18k 2    6   6c  15 18k 2   2  6 1  6c   2 1 1  tanh     1  c  k 2 , x 2 kt c      1 0   19 2    

Ac ce p

4. Conclusions

te

As case 4 we obtain solitary wave solutions E 4,2 and N 4,2 for Eq. (1) from Eq. (11).

These results shows that the simplest equation method is quite efficient and practically well suited for use in finding new traveling wave solutions for the coupled Schrodinger–Boussinesq equation. We can see that the technique used in this paper is very effective and can be steadily applied to nonlinear problems. On the other hand, it can be applied some nonintegrable equations, arising in applied mathematics.

References:

[1] Y.C. Hon, E.G. Fan, A series of exact solutions for coupled Higgs field equation and coupled Schrodinger–Boussinesq equation, Nonlinear Anal. 71 (2009) 3501–3508. [2] J. Lee, R. Sakthivel, Exact traveling wave solutions for some important nonlinear physical models, Pramana J. Phys. 80 (2013) 757–769. [3] J. Lee, R. Sakthivel, Exact travelling wave solutions of Schamel– Korteweg-de Vries equation, Rep. Math. Phys. 68 (2011) 153–161.

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[4] A. Bekir, A. Boz, Exact solutions for nonlinear evolution equations using Exp-function method, Phys. Lett. A 372 (2008) 1619–1625. [5] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons Fract. 30 (2006) 700–708.

ip t

[6] M.A. Akbar, N.H.M. Ali, Exp-function method for doffing equation and new solutions of (2+1)dimensional dispersive long wave equations, Prog. Appl. Math. 1 (2011) 30–42. [7] M. Wang, X. Li, J. Zhang, The (G0/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 (2008) 417–423.

cr

[8] H. Kim, R. Sakthivel, New exact travelling wave solutions of some nonlinear higher dimensional physical models, Rep. Math. Phys. 70 (2012) 39–50.

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[9] K. Khan, M.A. Akbar, Traveling wave solutions of nonlinear evolution equations via the enhanced (G0=G)-expansion method, J. Egypt. Math. Soc. (2013) (in press, http://dx.doi.org/10.1016/ j.joems.2013.07.009).

an

[10] M.A. Akbar, N.H.M. Ali, S.T. Mohyud-Din, The alternative (G0/G)-expansion method with generalized Riccati equation: application to fifth order (1+1)-dimensional Caudrey–Dodd– Gibbon equation, Int. J. Phys. Sci. 7 (2012) 743–752.

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[11] G. Ebadi, N.Y. Fard, H. Triki, A. Biswas, Exact solutions of the (2+1)-dimensional Camassa– Holm Kadomtsev–Petviashvili equation, Nonlinear Anal.: Model. Contr. 17 (2012) 280–296.

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[12] M. Eslami, M. Mirzazadeh, B. Fathi Vajargah, Anjan Biswas., Optical solitons for the resonant nonlinear Schrödinger's equation with time-dependent coefficients by the first integral method, Optik - International Journal for Light and Electron Optics, Volume 125, Issue 13, July 2014, Pages 3107-3116

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[13] M. Eslami, B. Fathi vajargah, M. Mirzazadeh., Exact solutions of modified Zakharov– Kuznetsov equation by the homogeneous balance method., Ain Shams Engineering Journal, Volume 5, Issue 1, March 2014, Pages 221-225. [14] M. Eslami, M. Mirzazadeh., First integral method to look for exact solutions of a variety of Boussinesq-like equations, Ocean Engineering, Volume 83, 1 June 2014, Pages 133-137. [15] M. Mirzazadeh, M. Eslami, A. Biswas, Soliton solutions of the generalized KleinGordon equation by using (G'/G )-expansion method, Comp. Appl. Math. DOI 10.1007/s40314-013-0098-3 [16] M. Eslami, M. Mirzazadeh, A. Biswas, Soliton solutions of the resonant non- linear Schrodinger's equation in optical _bers with time-dependent coe_cients by simplest equation approach, Journal of Modern Optics, 60(19) (2013) 16271636 [17] M. Eslami, M. Mirzazadeh, Topological 1-soliton solution of nonlinear Schrodinger equation with dual-power law nonlinearity in nonlinear optical fibers, Eur. Phys. J. Plus (2013) 128: 140. [18] M. Mirzazadeh, M. Eslami, Exact solutions for nonlinear variants of Kadomtsev-Petviashvili (n, n) equation using functional variable method, Pramana J. Phys. 81 (2013) 225-236.

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[12] Girifalco, L.A.: Statistical Physics of Materials. Wiley, New York (1973) Mir, Moscow (1975).

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[13] Kudryashov, N.A., Chaos SolitonFract. 24 (5) (2005) 1217–1231.

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