Exact solitons in optical metamaterials with quadratic-cubic nonlinearity using two integration approaches

Optik 156 (2018) 351–355

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Original research article

Exact solitons in optical metamaterials with quadratic-cubic nonlinearity using two integration approaches Mehmet Ekici Department of Mathematics, Faculty of Science and Arts, Bozok University, 66100 Yozgat, Turkey

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 12 September 2017 Accepted 8 November 2017

The aim of this paper is to present soliton solutions in optical metamaterials. The quadratic-cubic nonlinearity is considered. Two efficient algorithms that are the exp(−())-expansion method and extended Jacobi’s elliptic function expansion scheme are used to carry out the mathematical analysis. As a result, analytical dark soliton, singular soliton and periodic solutions are obtained. © 2017 Elsevier GmbH. All rights reserved.

Keywords: Solitons Metamaterials Quadratic-cubic nonlinearity

1. Introduction Metamaterials (MMs) are a class of new type of artificial synthetic materials with some extraordinary electromagnetic (EM) properties not existing in ordinary materials [1–4]. Recently, the study of optical solitons in MMs has attracted many researchers attention because of soliton theory in optical MMs is a very important and fascinating area of research in nonlinear optics. Zhou et al. [5] studied solitons in MMs with parabolic law nonlinearity. Xu et al. [6] reported Raman solitons in MMs having polynomial law non-linearity employing travelling wave hypothesis. Veljkovic et al. [7] studied super-sech soliton dynamics in MMs by collective variable approach. Triki et al. [8] investigated the MMs with Kerr law nonlinearity, and derived dipole soliton solutions by adopting the complex amplitude ansatz. Biswas et al. [9] obtained bright and dark solitons for MMs. Ebadi et al. [10] demonstrated the existence of solitons in MMs with Kerr law nonlinearity using F−expansion approach. In our most recent work [11], we analyzed the MMs with quadratic-cubic nonlinearity, and then acquired bright and singular optical soliton solutions by extended trial scheme approach. The nonlinear Schrödinger’s equation (NLSE) that governs the dynamics of soliton propagation through optical metamaterials with quadratic-cubic nonlinearity is studied in this paper, which in the dimensionless form is given by [9,12–15]











iqt + aqxx + b1 |q| + b2 |q|2 q = i ˛qx + ˇ |q|2 q



x

+  |q|2

  x





q + 1 |q|2 q

xx

+ 2 |q|2 qxx + 3 q2 q∗xx .

(1)

In this model, the complex valued dependent variable that represents the wave envelope is denoted by q and its complex conjugate is q*. The independent variables are x and t which respectively represent the spatial and temporal variables. The parameter a is the group velocity dispersion (GVD) while b1 and b2 together comprise the quadratic-cubic nonlinearity. On the right hand side, ˇ represents the self-steepening (SS) and  is the nonlinear dispersion (ND), while ˛ represents the inter-modal dispersion (IMD). Finally, the coefficients  l for l = 1, 2, 3 arise in the context of MMs [9,12–15]. This paper will perform the exp(−())-expansion approach [17–20] and extended Jacobi’s elliptic function expansion scheme [21–25] to obtain analytical solutions to the NLSE with quadratic-cubic nonlinearity. Details will be shown in the next sections.

E-mail address: [email protected] https://doi.org/10.1016/j.ijleo.2017.11.056 0030-4026/© 2017 Elsevier GmbH. All rights reserved.

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M. Ekici / Optik 156 (2018) 351–355

2. Soliton solutions To derive soliton solutions to the governing equation, the starting hypothesis is [9,13,16] q(x, t) = P() exp[i(x, t)],

(2)

where  = k(x − vt),

(3)

and the phase component  is given by (x, t) = −x + ωt + .

(4)

In (2) and (3), P(x, t) represents the amplitude portion of the soliton, and k and v are inverse width and velocity of soliton. The parameters  and ω in (4) represent the frequency and wave number of the soliton, respectively while  is the phase constant. Substituting (2) into (1), one obtains a pair of relations. Imaginary part gives

v = −˛ − 2a,

(5)

3ˇ + 2 − 2(31 + 2 − 3 ) = 0,

(6)

and

while real part leads to









ak2 P  − ω + a2 + ˛ P + b1 P 2 + b2 − ˇ + 2 1 + 2 2 + 2 3 P 3





− k2 31 + 2 + 3 P 2 P  − 6k2 1 P(P  )2 = 0.

(7)

In order to extract an analytic solution, we apply the transformations  1 = 0 and  2 =−  3 in Eq. (7) to find





ak2 P  − ω + a2 + ˛ P + b1 P 2 + (b2 − ˇ)P 3 = 0,

(8)

where 3ˇ + 2 + 43 = 0.

(9)

The exp(−())-expansion approach and extended Jacobi’s elliptic function expansion scheme will now be applied, in the following sections, to Eq. (8) to obtain bright, dark and singular soliton solutions to (1). 3. exp(−())-Expansion approach To start off with exp(−())-expansion approach, the initial assumption of the solution structure of (8) is taken to be: P() =

N 

i

Ai (exp[−()]) ,

(10)

i=1

where Ai for i = 0, 1, . . ., N are constants to be determined later, such that AN = / 0, while the function () is the solution of the auxiliary ordinary differential equation (ODE)  () = exp[−()] + exp[()] + .

(11)

It is well known that Eq. (11) has solutions in the following forms: If = / 0 and 2 − 4 > 0,

√ 2  ⎛ ⎞

−4

2 − 4 tanh ( + C) +

2 ⎠. () = ln ⎝− 2 For = / 0 and 2 − 4 < 0,

⎛

() = ln ⎝

4 − 2 tan

√



4 − 2 ( + C) 2

2

−

(12)

⎞ ⎠.

(13)

However, when = 0, = / 0 and 2 − 4 > 0, () = − ln



exp( ( + C)) − 1



.

(14)

M. Ekici / Optik 156 (2018) 351–355

353

Whenever = / 0, = / 0 and 2 − 4 = 0,



() = ln

2( ( + C) + 2) −

2 ( + C)



.

(15)

Finally, if = 0, = 0 and 2 − 4 = 0, () = ln( + C).

(16) P

It should be noted that C is the integration constant. Balancing with expansion scheme suggests the use of the finite expansion given by

P3

in Eq. (8) gives N = 1. Therefore, the exp(−())-

P() = A0 + A1 exp[−()].

(17)

Substituting (17) along with (11) into Eq. (8) and equating all the coefficients of powers of exp(−()) to be zero, one obtains a system of algebraic equations. Solving this system by Mathematica yields



A0 =



3 ab1 k2 ( 2 − 4 ) ± H

, A1 = ±

2b21



3H b21

, (18)



ω = −˛ − a 2 − k2 ( 2 − 4 ) , b2 = ˇ −

2b21 9ak2 ( 2 − 4 )

,

where and are arbitrary constants, and H is given by



H=

a2 b21 k4 2 ( 2 − 4 ).

(19)

Substituting the solution set (18) into (17), the solution formula of Eq. (8) can be written in the form:



P() =



3 ab1 k2 ( 2 − 4 ) ± H 2b21

±

3H b21

exp[−()].

(20)

As a consequence, one recovers exact solutions to the governing equation as follows: Substituting the solution given by (12) of auxiliary ODE into (20), it is obtained hyperbolic function solutions as

⎧ ⎛ ⎞⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟⎪ ⎪ ⎪ ⎬ ⎨ 3 ab k2 ( 2 − 4 ) ± H ⎜ ⎟ 1 2 3H ⎜ ⎟   q(x, t) = ∓ ⎟⎪ 2b21 b21 ⎜ ⎪

2 − 4 ⎪ ⎝ 2 ⎠⎪ ⎪ ⎪ ⎪ ⎪

− 4 tanh + C ) +

( ⎭ ⎩ 2    2 2 2   × exp i −x + −˛ − a  − k ( − 4 )

t+

(21)

.

Substituting the solution given by (13) of auxiliary ODE into (20), it is recovered trigonometric function solutions as

⎧ ⎛ ⎞⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟⎪ ⎪ ⎬ ⎨ 3 ab k2 ( 2 − 4 ) ± H ⎜ ⎟ 1 2 3H ⎜ ⎟   q(x, t) = ± ⎟⎪ b21 ⎜ 2b21 ⎪ 4 − 2 ⎪ ⎝ ⎠⎪ ⎪ ⎪ 2 tan ⎪ ⎪ 4 −

+ C) −

( ⎭ ⎩ 2    2 2 2   × exp i −x + −˛ − a  − k ( − 4 )

t+

(22)

.

Substituting the solution given by (14) of auxiliary ODE into (20), it is found hyperbolic function solutions as

 



3 ab1 k2 ( 2 − 4 ) ± H

q(x, t) =



2b21



× exp i −x + −˛ − a

±



2

3H b21





exp( ( + C)) − 1

− k2 ( 2



− 4 )

t+







(23) .

4. Extended Jacobi’s elliptic function expansion scheme Suppose that the structure solution of (8) is given by P() =

N  j=0

˛j snj  +

N  j=1

ˇj sn−j ,

(24)

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M. Ekici / Optik 156 (2018) 351–355

where ˛0 , ˛j and ˇj for j = 1, 2, . . ., N are constants to be determined later. Balancing P with P3 in Eq. (8), we find that N = 1. Then, the solution has the form P() = ˛0 + ˛1 sn  + ˇ1 sn−1 .

(25)

Substituting (25) into (8) and equating all the coefficients of powers of sn  to be zero, then one obtains a system of nonlinear algebraic equations and by solving it, one recovers Set-1.



ˇ1 = 0, ˛0 = ω=

3ak2 m2 2(1 + m2 ) 3ak2 (1 + m2 ) , ˛1 = ± , b1 b1

2ak2 (1 + m2 ) − ˛

− a2 ,

b21

b2 = ˇ −

9ak2 (1 + m2 )

Set-2. ˛1 = 0, ˛0 = ω=

3ak2 3ak2 (1 + m2 ) , ˇ1 = ± b1

2ak2 (1 + m2 ) − ˛

− a2 ,

Set-3.

2(1 + m2 )

,

b1

(27)

9ak2 (1 + m2 )

,



3ak2 m 3ak2 (1 + m(6 + m)) ˛0 = , ˛1 = ∓ b1 ω=

,



b21

b2 = ˇ −

(26)

2ak2 (1 + m(6 + m)) − ˛

− a2 ,

2(1 + m(6 + m)) b1

b2 = ˇ −

, ˇ1 = ∓

b21 9ak2 (1 + m(6 + m))

3ak2



2(1 + m(6 + m)) b1

, (28)

.

Thus, we acquire the following Jacobi elliptic function solutions to the governing model:



q(x, t) =





3ak2 m2 2(1 + m2 ) 3ak2 (1 + m2 ) ± sn[k(x + (˛ + 2a)t)] b1 b1





× exp i −x + 2ak2 (1 + m2 ) − ˛ − a

 q(x, t) =

3ak2 3ak2 (1 + m2 ) ± b1



× exp i −x +

 q(x, t) =





2(1 + m2 ) b1

t+



− a2



t+



 (30)

,



2(1 + m(6 + m)) b1

× (m sn[k(x + (˛ + 2a)t)] + ns[k(x + (˛ + 2a)t)])



,

(29)

ns[k(x + (˛ + 2a)t)]

2ak2 (1 + m2 ) − ˛

3ak2 3ak2 (1 + m(6 + m)) ∓ b1

 2





(31)



× exp i −x + 2ak2 (1 + m(6 + m)) − ˛ − a2 t + 



.

When the modulus m → 1 in Eqs. (29)–(31), solitary wave solutions emerge

!

















q(x, t) =

6ak2 (1 ± tanh[k(x + (˛ + 2a)t)]) b1

exp i −x + 4ak2 − ˛ − a2 t + 

q(x, t) =

6ak2 (1 ± coth[k(x + (˛ + 2a)t)]) b1

exp i −x + 4ak2 − ˛ − a2 t + 

q(x, t) =

24ak2 (1 ∓ coth 2[k(x + (˛ + 2a)t)]) b1

! !







,

(32)

,

(33)

exp i −x + 16ak2 − ˛ − a2 t + 



.

(34)

However, if m → 0 in Eqs. (30) and (31), trigonometric function solutions are obtained q(x, t) =

 √ 3ak2  1 ± 2csc[k(x + (˛ + 2a)t)] b1

q(x, t) =

 √ 3ak2  1 ∓ 2csc[k(x + (˛ + 2a)t)] b1

!

















exp i −x + 2ak2 − ˛ − a2 t + 

!

exp i −x + 2ak2 − ˛ − a2 t + 

,

(35)

.

(36)

M. Ekici / Optik 156 (2018) 351–355

355

5. Conclusions The perturbed NLSE describing the dynamics of optical solitons in MMs, in the presence of quadratic-cubic nonlinearity, IMD, SS as well as ND, has been studied analytically in this paper. There are two types of integration technique that are adopted. They are the exp(−())-expansion approach and extended Jacobi’s elliptic function expansion scheme. The first algorithm retrieved some hyperbolic and trigonometric function solutions. The extended Jacobi’s elliptic function expansion approach retrieved Jacobian elliptic periodic traveling wave solutions to the model. In the limiting situations, the elliptic solutions obtained by this integration technique are reduced to solitary waves, shock waves and singular solitons. References [1] D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84 (18) (2000) 4184. [2] W. Cui, Y. Zhu, H. Li, S. 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