Exact solitons to generalized resonant dispersive nonlinear Schrödinger's equation with power law nonlinearity

Optik 130 (2017) 178–183

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Original research article

Exact solitons to generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity M. Mirzazadeh a,∗ , Mehmet Ekici b , Qin Zhou c , Anjan Biswas d,e a Department of Mathematics, Faculty of Mathematical Sciences, East of Guilan, University of Guilan, P.C. 44891-63157 Rudsar-Vajargah, Iran b Department of Mathematics, Faculty of Science and Arts, Bozok University, 66100 Yozgat, Turkey c School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, PR China d Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria-0008, South Africa e Faculty of Science, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 16 September 2016 Accepted 7 November 2016 Keywords: Bright solitons Singular periodic solutions Integrability

a b s t r a c t This work focuses on the solitons of the generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity. Based on the improved extended tanh-equation method, explicit bright solitons, singular periodic solutions, rational solutions, Weierstrass elliptic doubly periodic type solution and some other types solutions are derived. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction The generalized resonant dispersive nonlinear Schrödinger’s equation (GRD-NLSE) is an important mathematical-physical model in nonlinear science including the nonlinear optics, fluid mechanics and condensed matter physics [1–17]. Recently, many researchers presented the analytical study to this model, and constructed some exact solutions by many integration algorithms. In this work, we will use a different method that is the improved extended tanh-equation method to investigate the dynamics of solitons to this model. 2. Improved modified extended tanh-function method In this section, we describe the improved extended tanh-equation method [1] for finding traveling wave solutions of nonlinear partial differential equations (NLPDE) and subsequently we will apply this method to solve the GRD-NLSE with power law nonlinearity. We suppose that the given NLPDE for u(x, t) is in the form P (u, ut , ux , uxx , uxt , utt , . . .) = 0,

(1)

where P is a polynomial. The essence of the improved extended tanh-equation method can be presented in the following steps:

∗ Corresponding author. E-mail address: [email protected] (M. Mirzazadeh). http://dx.doi.org/10.1016/j.ijleo.2016.11.036 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

M. Mirzazadeh et al. / Optik 130 (2017) 178–183

179

Step-1: To find the traveling wave solutions of Eq. (1), we introduce the wave variable u(x, t) = U(),

 = x − vt,

(2)

where v is a constant to be determined later. Substituting Eq. (2) into Eq. (1), we obtain the following ODE





Q U, U  , U  , . . . = 0.

(3)

Step-2: The basic idea of the improved extended tanh-equation method [1] consists in expanding the solutions U() of Eq. (3) in a finite series U() =

N 

al z l +

N 

bl z −l

(4)

c0 + c1 z + c2 z 2 + c3 z 3 + c4 z 4 ,

(5)

l=0

l=1

where z = z() satisfies



z = ε

where ε =±1 . Eq. (5) gives various kinds of fundamental solutions [1]. From these solutions, more new exact solutions for Eq. (1) can be obtained. Step-3: Determine the positive integer number N in Eq. (4) by balancing the highest order derivatives and the nonlinear terms in Eq. (3) Step-4: Substitute Eq. (4) along with (5) into Eq. (3), we obtain a polynomial of z. Equating the coefficients of this polynomial to zero, we get a system of algebraic equations which can be solved by theMaple to get the unknown parameters v, a0 , al and bl (l = 1, b, . . .). As a result, we obtain the exact solutions to Eq. (1). 3. GRD-NLSE with power law nonlinearity The model equation that will be studied in this paper is given by the GRD-NLSE that reads [2–9]



i | |n−1





t

+ ˛ | |n−1



 xx

+ ˇ| |m

+

(| |n )xx | |



= 0.

(6)

For this model, (x, t) is the wave profile and represents the complex valued function. ˛ and ˇ respectively represent the coefficients of the generalized group velocity dispersion (GVD) and power law nonlinearity. Then  represent the coefficient of the resonant term that appears in the study of Madelung fluids. The parameter m represents power law nonlinearity. When m = 2, this model equation collapses to Kerr law nonlinearity that is also referred to cubic NLSE. Finally, the parameter n dictates the generalized evolution and generalized GVD. For n = 1, this model equation collapses to the regular NLSE. This parameter n thus maintains the evolution and GVD on a generalized setting. During long distance soliton propagation, the evolution and the GVD gets distorted and modified. Therefore, it is imperative to consider NLSE where the evolution and GVD are modified to maintain the dynamics of soliton propagation on a closer to reality setting. Therefore this GRD-NLSE is the proposed model. Under the travelling wave transformation (x, t) = U()ei(−x+ωt+) , we have







 = x + 2˛t,

(7)



(˛ + ) U n  − ω + 2 ˛ U n + ˇU m+1 = 0,

(8)

In order to obtain closed form solutions, we use the transformation U() = V 1/m+1−n ,

(9)

that will reduce Eq. (8) into the ODE

 2

(˛ + ) n(2n − m − 1) V 





+ (˛ + ) n(m + 1 − n)VV  − ω + 2 ˛ (m + 1 − n)2 V 2 + ˇ(m + 1 − n)2 V 3 = 0.

(10)

3.1. Application of improved modified extended tanh-function method We will now analyze Eq. (10) to obtain soliton solutions by means of improve the modified extended tanh-function method. According to the homogeneous balance method, Eq. (10) has the solution in the form V () = a0 + a1 z + a2 z 2 +

b2 b1 + 2, z z

(11)

Substituting (11) along with Eq. (5) into Eq. (10), collecting the coefficients of z, and solving the resulting system with the aid of theMaple, we find the following results:

180

M. Mirzazadeh et al. / Optik 130 (2017) 178–183

Case 1.

c0 = c1 = c3 = 0 . We have

a0 = a1 = b1 = b2 = 0,

a2 = −

2nc 4 (n + m + 1) (˛ + ) ˇ(m + 1 − n)2



,



(m + 1 − n)2 ω + 2 ˛

c2 =

4n2 (˛ + )

.

(12)

We obtain bright soliton, singular periodic and rational solutions (1) Bright soliton solution

⎧ ⎨ (n + m + 1) ω + 2 ˛

(x, t) =

⎩

⎡ sech2 ⎣

2nˇ





(m + 1 − n)2 ω + 2 ˛ 4n2 (˛ + )

⎤⎫1/m+1−n ⎬ ei{−x+ωt+} . (x + 2˛t)⎦ ⎭

(13)

This soliton is valid for





ω + 2 ˛ (˛ + ) > 0.

(2) Periodic singular solution

⎧ ⎨ (n + m + 1) ω + 2 ˛

(x, t) =

⎩

⎡ sec2 ⎣

2nˇ

−





(m + 1 − n)2 ω + 2 ˛ 4n2 (˛ + )

⎤⎫1/m+1−n ⎬ ei{−x+ωt+} . (x + 2˛t)⎦ ⎭

(14)

Eq. (14) is valid when





ω + 2 ˛ (˛ + ) < 0.

(3) Rational solution

 (x, t) = Case 2. (I):

−

1/m+1−n 

2n(n + m + 1) (˛ + )

1

ˇ(m + 1 − n)2

(x + 2˛t)2

e

i −x−˛2 t+

 .

(15)

c1 = c3 = 0, c0 = c22 /4c4 . We have

a0 = −

nc 2 (m + n + 1)(˛ + ) ˇ(m + 1 − n)

2

,

a1 = a2 = b1 = 0,

b2 = −

nc 22 (n + m + 1) (˛ + ) 2ˇc4 (m + 1 − n)

2

,

ω = −2 ˛ −

2n2 c2 (˛ + ) (m + 1 − n)2

. (16)

We obtain singular soliton and singular periodic solutions

 (x, t) =

nc 2 (m + n + 1)(˛ + ) ˇ(m + 1 − n)2

2

csch



c2 − (x + 2˛t) 2

1/m+1−n  e





i −x− 2 ˛+(2n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(17)

This soliton is valid for c2 < 0.

 (x, t) =

−

nc 2 (m + n + 1)(˛ + ) ˇ(m + 1 − n)2

 csc2

c2 (x + 2˛t) 2

1/m+1−n  e





i −x− 2 ˛+(2n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(18)

Eq. (18) is valid when c2 > 0. (II): a0 = −

nc 2 (m + n + 1)(˛ + ) ˇ(m + 1 − n)

2

, a1 = b1 = b2 = 0,

a2 = −

2nc 4 (n + m + 1) (˛ + ) ˇ(m + 1 − n)

2

,

ω = −2 ˛ −

2n2 c2 (˛ + ) (m + 1 − n)2

. (19)

M. Mirzazadeh et al. / Optik 130 (2017) 178–183

181

We obtain bright soliton, singular periodic solutions

 (x, t) =

−

nc 2 (m + n + 1)(˛ + )

sech2

ˇ(m + 1 − n)2

 −

c2 (x + 2˛t) 2

1/m+1−n  e





i −x− 2 ˛+(2n2 c2 (˛+)/(m+1−n)2 ) t+

 . (20)

This soliton is valid for c2 < 0.

 (x, t) =

−



nc 2 (m + n + 1)(˛ + )

sec

ˇ(m + 1 − n)2

c2 (x + 2˛t) 2

2

1/m+1−n  e





i −x− 2 ˛+(2n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(21)

Eq. (21) is valid when c2 > 0. (III): a0 = −

b2 = −

2nc 2 (m + n + 1)(˛ + )

,

a1 = b1 = 0,

,

ω = −2 ˛ −

ˇ(m + 1 − n)2 nc 22 (n + m + 1) (˛ + ) 2ˇc4 (m + 1 − n)

2

a2 = −

2nc 4 (n + m + 1) (˛ + ) ˇ(m + 1 − n)2

2n2 c2 (˛ + ) (m + 1 − n)2

.

(22)

We get soliton, singular periodic solutions

 (x, t) =

−

nc 2 (m + n + 1)(˛ + )

(x, t) =



−

nc 2 (m + n + 1)(˛ + )

Case 3.

 2 + tan





c2 2 (x + 2˛t) − coth 2

 −

c2 (x + 2˛t) 2

1/m+1−n

.

2

i −x− 2 ˛+(2n2 c2 (˛+)/(m+1−n)2 ) t+



−





ˇ(m + 1 − n)2



×e



i −x− 2 ˛+(2n2 c2 (˛+)/(m+1−n)2 ) t+





2 − tanh2

ˇ(m + 1 − n)2



×e



(23)



c2 (x + 2˛t) + cot2 2



c2 (x + 2˛t) 2

1/m+1−n

 .

(24)

c0 = c1 = c4 = 0 . We have

a0 = a2 = b1 = b2 = 0,

a1 = −

nc 3 (n + m + 1) (˛ + ) 2ˇ(m + 1 − n)

2

, ω = −2 ˛ +

n2 c2 (˛ + ) (m + 1 − n)2

.

(25)

We obtain bright soliton, singular periodic and rational solutions

 (x, t) =

nc 2 (m + n + 1)(˛ + ) 2ˇ(m + 1 − n)2

2

sech

 √c

2

2

(x + 2˛t)

1/m+1−n  e





i −x− 2 ˛−(n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(26)

This soliton is valid for c2 > 0.

 (x, t) =

nc 2 (m + n + 1)(˛ + ) 2ˇ(m + 1 − n)2

sec2

 √−c

2

2

(x + 2˛t)

1/m+1−n  e





i −x− 2 ˛−(n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(27)

Eq. (27) is valid when c2 < 0.

 (x, t) =

−

2n(n + m + 1) (˛ + )

1

ˇ(m + 1 − n)2

(x + 2˛t)2

1/m+1−n  e

i −x−˛2 t+

 .

(28)

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M. Mirzazadeh et al. / Optik 130 (2017) 178–183

Case 4.

c0 = / 0,

c2 = c4 = 0,

a2 = b1 = b2 = 0,

c1 = / 0, nc 21

a0 =

c3 > 0 . We get

(n + m + 1) (˛ + )

9c0 ˇ(m + 1 − n)

2

,

a1 =

2nc 31 (n + m + 1) (˛ + ) 27c02 ˇ(m + 1 − n)2

,

2n2 c12 (˛ + )

ω = −2 ˛ +

, 4c13 + 27c02 c3 = 0. 3c0 (m + 1 − n)2 We obtain Weierstrass elliptic doubly periodic type solution

 (x, t) =

nc 21 (n + m + 1) (˛ + )

2c1 1+ ℘ 3c0

9c0 ˇ(m + 1 − n)2



×e







−



i −x− 2 ˛−(2n2 c 2 (˛+)3c0 /(m+1−n)2 ) t+



1

(29)

c13 27c02

 !1/m−n+1 (x + 2˛t) , g2 , g3

,

(30)

where g2 =

27c02 c12

,

g3 =

27c03 c13

,

(31)

are called invariants of the Weierstrass elliptic function. Case 5. (I):

c3 = c4 = 0 . We have

a0 = a1 = a2 = 0, ω = −2 ˛ +

b1 = −

n2 c2 (˛ + ) (m + 1 − n)

2

,

nc 1 (n + m + 1) (˛ + ) ˇ(m + 1 − n)2 c0 =

c12 4c2

,

b2 = −

nc 21 (n + m + 1) (˛ + ) 2c2 ˇ(m + 1 − n)2

,

.

(32)

We obtain an exponential type solution

 (x, t) =

−

4nc 1 c22 (n + m + 1) (˛ + ) ˇ(m + 1 − n)2



×e



.

1/m−n+1 √ exp [ε c2 (x + 2˛t)] √ 2 (c1 − 2c2 exp [ε c2 (x + 2˛t)]) 

i −x− 2 ˛−(n2 c2 (˛+)/(m+1−n)2 ) t+



.

(33)

Eq. (33) is valid when c2 > 0. (II): a0 = a1 = a2 = b2 = 0,

b1 = −

nc 1 (n + m + 1) (˛ + ) 2ˇ(m + 1 − n)

2

,

ω = −2 ˛ +

n2 c2 (˛ + ) (m + 1 − n)2

,

c0 = 0.

(34)

We have solitary wave solution and triangular type solution

 (x, t) =

−

c2 n (n + m + 1) (˛ + ) ˇ(m + 1 − n)2

1 √ −1 + ε sinh [2 c2 (x + 2˛t)]

1/m−n+1



×e





i −x− 2 ˛−(n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(35)

This soliton is valid for c2 > 0.

 (x, t) =

−

c2 n (n + m + 1) (˛ + ) ˇ(m + 1 − n)2

Eq. (36) is valid when c2 < 0.

1 √ −1 + ε sin [ −c2 (x + 2˛t)]

1/m−n+1



×e





i −x− 2 ˛−(n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(36)

M. Mirzazadeh et al. / Optik 130 (2017) 178–183

c0 = c1 = 0, c4 > 0 . We have √ 2n c2 c4 (n + m + 1) (˛ + ) a0 = b1 = b2 = 0, a1 = , ˇ(m + 1 − n)2

183

Case 6.

a2 = −

√ , c3 = −2 c2 c4 . (m + 1 − n)2 We obtain bright soliton solution ω = −2 ˛ +



(x, t) =

2nc 4 (n + m + 1) (˛ + ) 2c2 ˇ(m + 1 − n)2

,

n2 c2 (˛ + )

nc 2 (m + n + 1)(˛ + ) 2ˇ(m + 1 − n)2

2

sech

(37)

 √c

2

2

(x + 2˛t)

1/m+1−n  e





i −x− 2 ˛−(n2 c2 (˛+)/(m+1−n)2 ) t+

 .

(38)

This soliton is valid for c2 > 0. 4. Conclusions The GRD-NLSE with power law nonlinearity has been studied analytically. The mathematical method that is the improved extended tanh-equation method is applied to extract exact solitons. As a result, some soliton solutions are reported. Acknowledgements This work was supported by the Program for Outstanding Young and Middle-aged Scientific and Technological Innovation Team of the Higher Education Institutions of Hubei Province of China under the grant number T201525. References [1] Zonghang Yang, Benny Y.C. Hon, An improved modified extended tanh-function method, Z. Naturforsch. 61a (2006) 103–115. [2] H. Triki, T. Hayat, O.M. Aldossary, A. Biswas, Bright and dark solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients, Opt. Laser Technol. 44 (2012) 2223–2231. [3] H. Triki, A. Yildirim, T. Hayat, Omar M. Aldossary, A. Biswas, 1-Soliton solution of the generalized resonant nonliner dispersiv Schrödinger’s equation with time-dependent coefficients, Adv. Sci. Lett. 16 (2012) 309–312. [4] A. Biswas, Soliton solutions of the perturbed resonant nonliner dispersiv Schrödinger’s equation with full nonlinearity by semi-inverse variational principle, Quant. Phys. Lett. 1 (2) (2012) 79–84. [5] M. Eslami, M. Mirzazadeh, A. Biswas, Soliton solutions of the resonant nonlinear Schrödinger’s equation in optical fibers with time dependent coefficients by simplest equation approach, J. Mod. Opt. 60 (19) (2013) 1627–1636. [6] M. Mirzazadeh, M. Eslami, Daniela Milovic, A. Biswas, Topological solitons of resonant nonliner Schrödinger’s equation with dual-power law nonlinerity using G’/G-expansion technique, Optik 125 (19) (2014) 5480–5489. [7] M. Eslami, M. Mirzazadeh, B. Fathi Vajargah, A. Biswas, Optical solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients by the first integral method, Optik 125 (9) (2014) 3107–3116. [8] M. Mirzazadeh, M. Eslami, B. Fathi Vajargah, A. Biswas, Optical solitons and optical rogons of generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity, Optik 125 (9) (2014) 4246–4256. [9] M. Mirzazadeh, A.H. Arnous, M.F. Mahmood, E. Zerrad, A. Biswas, Soliton solutions to resonant nonlinear Schrödinger’s equation with time-dependent coefficients by trial solution approach, Nonlinear Dyn. 81 (1–2) (2015) 277–282. [10] C.Q. Dai, Y.Y. Wang, Spatiotemporal localizations in (3+1)-dimensional PT-symmetric and strongly nonlocal nonlinear media, Nonlinear Dyn. 83 (2016) 2453–2459. [11] R. Guo, Y.F. Liu, H.Q. Hao, F.H. Qi, Coherently coupled solitons, breathers and rogue waves for polarized optical waves in an isotropic medium, Nonlinear Dyn. 80 (2015) 1221–1230. [12] A.M. Wazwaz, S.A. El-Tantawy, A new (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation, Nonlinear Dyn. 84 (2) (2016) 1107–1112. [13] X. Lü, Madelung fluid description on a generalized mixednonlinear Schrödinger equation, Nonlinear Dyn. 81 (2015) 239–247. [14] Q. Zhou, Q. Zhu, H. Yu, X. Xiong, Optical solitons in media with time-modulated nonlinearities and spatiotemporal dispersion, Nonlinear Dyn. 80 (2015) 983–987. [15] Q. Zhou, L. Liu, Y. Liu, H. Yu, P. Yao, C. Wei, H. Zhang, Exact optical solitons in metamaterials with cubic-quintic nonlinearity and third-order dispersion, Nonlinear Dyn. 80 (2015). [16] W.J. Liu, L. Huang, P. Huang, Y. Li, M. Lei, Dark soliton control in inhomogeneous optical fibers, Appl. Math. Lett. 61 (2016) 80–87. [17] Q. Zhou, Q. Zhu, H. Yu, Y. Liu, C. Wei, P. Yao, A.H. Bhrawy, A. Biswas, Bright, dark and singular optical solitons in a cascaded system, Laser Phys. 25 (2015) 025402.