Optical solitons to Kundu–Mukherjee–Naskar model with modified simple equation approach

Optik - International Journal for Light and Electron Optics 184 (2019) 247–252

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Original research article

Optical solitons to Kundu–Mukherjee–Naskar model with modified simple equation approach

T

Yakup Yıldırım Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey

A R T IC LE I N F O

ABS TRA CT

Keywords: Modified simple equation procedure Kundu–Mukherjee–Naskar model Solitons

The Kundu–Mukherjee–Naskar model is examined to uncover quite important optical solitons in (2 + 1)-dimensions of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

1. Introduction Solitons in optics are molecules or pulses that compose the basic fabric of soliton transmission technology in data transmissionacross, transoceanic and transcontinental distances, telecommunications and also optical fibers over the globe in a matter of a few femto-seconds. There are several models that describe this phenomena. They are Kaup–Newell equation, Lakshmanan–Porsezian–Daniel model, Gerdjikov–Ivanov equation, Schrödinger–Hirota model, complex Ginzburg–Landau equation, Radhakrishnan–Kundu–Lakshmanan equation, Fokas–Lenells equation and many more [4–13] where soliton pulses in (1 + 1)-dimensions have been taken into consideration. Kundu–Mukherjee–Naskar model also is one of these quite important models and moreover has been utilized in prominent procedures. The most important feature of this governing model is that the model has been given as a new extension of nonlinear Schrödinger equation with the inclusion of different form nonlinearity with respect to Kerr and non-Kerr law nonlinearities to work soliton pulses in (2 + 1)-dimensions. Another important aspect of this model is that this model is used to govern optical wave propagation along with coherently excited resonant waveguides in particular in the phenomena of bending of light beams [1]. Soliton dynamics in (1 + 1)-dimensions were investigated by many researchers in [2–9] while solitons in (2+1)-dimensions were taken into account by just a few investigators in [1–3]. To contribute to the works done on the model so far, the Kundu–Mukherjee–Naskar model is taken into consideration in this manuscript using modified simple equation architecture that causes to obtain optical solitons in additional singular-periodic solutions with parameter restrictions. These conditions are important because they guarantee the existence of the solutions. The details of this very valuable work mentioned in this section is yield in the following chapters of this manuscript. 1.1. Governing model The Kundu–Mukherjee–Naskar model [1,2] is given as

iψt + aψxy + ibψ (ψψx* − ψ*ψx ) = 0.

(1)

The temporal evolution of pulses is ensured by the first term whilst the profile of the optical solitons is given by means of the function ψ (x , y, t ) . Moreover, the dispersion term and the nonlinearity term are assured by the coefficient of a, b respectively.

E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.02.135 Received 23 February 2019; Accepted 23 February 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 184 (2019) 247–252

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2. A quick glance over modified simple equation approach A quick glance over the modified simple equation architecture [4–13] is yield to study quite important solitons. Step-1: A nonlinear evolution equation can be given by

V (r , rt , rx , rtt, rxt, rxx , …) = 0

(2)

with the dependent function and its partial derivatives shown as r , rt , rx , rtt, rxt, rxx , … and also this equation is decreased in

F (P , P′, P′ ′, P′ ′′, …) = 0

(3)

by using of the conversion

r (x , t ) = P (s )

(4)

s = x − pt.

(5)

with

The dependent function and its derivatives are given by P , P′, P′ ′, P′ ′′, … sequentially in the ordinary differential equation (3). Step-2: The ancillary equation for this methodology is yield as N

P (s ) =

Q′ (s )

i

∑ δi ⎛ Q (s ) ⎞ i=0

⎜

⎟

⎝

⎠

(6)

having the essential constant coefficients δ0, δ1, …, δN . Step-3: The application of Eq. (6) is attached to the N number which can be obtained with the aid of balancing rule in the ordinary differential equation (3). Step-4: The overdeterminet equations are acquired with the aid of putting Eq. (6) in Eq. (3) and setting of the constant coefficients of the functions Q−l , l = 0, 1, 2… to zero. Thus, the explicit solutions to Eq. (2) are acquired if the requisite constants δ0, δ1, …, δN can be given with the aid of solving the equations. 2.1. Implementation to the model With a view to study quite important optical solitons with the Kundu–Mukherjee–Naskar model, the following transformation

ψ (x , y, t ) = P (ϑ) eiφ (x , y, t )

(7)

through the medium of

ϑ = B1 x + B2 y − ρt

(8)

will be considered. The function P (ϑ) means the amplitude component and the function φ (x , y, t ) signifies the phase component that can be supposed by

φ (x , y, t ) = −κ1 x − κ2 y + ωt + ζ .

(9)

The parameters κ1, κ2 mean soliton frequency and the parameter ω stands for soliton wave number whilst the parameter ζ signifies soliton phase. The imaginary part is recovered by

ρ = −aB1κ2 − aB2 κ1

(10)

while the real part is acquired as

aB1B2 P′ ′ − (ω + aκ1 κ2) P − 2bκ1 P 3 = 0

(11)

because of inserting Eq. (7) into Eq. (1). Case-1: Eq. (6) can be yield by

Q′ (ϑ) ⎞ P (ϑ) = δ0 + δ1 ⎛ ⎝ Q (ϑ) ⎠ ⎜

⎟

(12) ′

because of N = 1 which can be obtained by using of balancing rule P′ with The overdeterminet equations are acquired as follows Q−3 coeff.:

2δ1 (Q′)3 (−bδ12κ1 + aB1B2) = 0, Q−2

P3

in the ordinary differential equation (11).

(13)

coeff.:

− 3δ1 Q′ (2bδ0 δ1 κ1 Q′ + aB1B2 Q′ ′) = 0,

(14) 248

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Q−1 coeff.:

− δ1 ((6bδ0 2κ1 + aκ1 κ2 + ω) Q′ − aB1B2 Q′ ′′) = 0,

(15)

Q 0 coeff.: − 2bδ03κ1 − aδ0 κ1 κ2 − ωδ0 = 0,

(16)

because of putting Eq. (12) in Eq. (11) and setting of the constant coefficients of the functions The following results are given by

δ0 = ± −

aκ1 κ2 + ω , 2bκ1

δ1 = ±

aB1B2 bκ1

Q−3,

Q−2,

Q−1,

Q0

to zero sequentially.

(17)

and

Q′ ′ = ± −

Q′ ′′ = −

2(aκ1 κ2 + ω) Q′, aB1B2

(18)

2(aκ1 κ2 + ω) Q′. aB1B2

(19)

The following equations can be reached as

Q′ = ± −

aB1B2 ± k1 e 2(aκ1 κ2 + ω)

−

2(aκ1 κ2+ ω) ϑ aB1B2 ,

(20)

and

Q=−

aB1B2 ± k1 e 2(aκ1 κ2 + ω)

−

2(aκ1 κ2+ ω) ϑ aB1B2

+ k2,

(21)

with k1 and k2 integration constants by means of using Eqs. (18) and (19). The solitons of the Kundu–Mukherjee–Naskar model are emerged as follows

⎧ ⎪ aκ κ + ω ψ (x , y, t ) = ± − 12bκ2 ± 1 ⎨ ⎪ ⎩ × ei (−κ1 x − κ2 y + ωt + ζ )

aB1B2 bκ1

2(aκ κ + ω) ⎛ ± − aB1B2 k1 e± − aB11B22 (B1x + B2 y − ρt ) ⎞ ⎫ ⎪ 2(aκ1 κ2 + ω) ⎜ ⎟ 2(aκ κ + ω) ⎜ − aB1B2 k e± − aB11B22 (B1x + B2 y − ρt ) + k ⎟ ⎬ 2 ⎪ 1 ⎝ 2(aκ1κ2 + ω) ⎠⎭

(22)

on account of inserting Eqs. (20) and (21) into Eq. (12). If we set

k1 = −

2(aκ1 κ2 + ω) ± e aB1B2

−

2(aκ1 κ2+ ω) ϑ0 aB1B2 ,

k2 = ± 1,

(23)

we get:

ψ (x , y, t ) = ± −

aκ1 κ2 + ω aκ κ + ω tanh ⎡ − 1 2 (B1 x + B2 y − ρt + ϑ0) ⎤ ei (−κ1 x − κ2 y + ωt + ζ ). ⎢ ⎥ 2bκ1 2aB1B2 ⎣ ⎦

(24)

The solution (24) points out dark soliton provided that

aB1B2 (aκ1 κ2 + ω) < 0. ψ (x , y, t ) = ± −

aκ1 κ2 + ω aκ κ + ω coth ⎡ − 1 2 (B1 x + B2 y − ρt + ϑ0) ⎤ ei (−κ1 x − κ2 y + ωt + ζ ). ⎢ ⎥ 2bκ1 2aB1B2 ⎣ ⎦

(25)

The solution (25) points out singular soliton provided that

aB1B2 (aκ1 κ2 + ω) < 0. ψ (x , y, t ) = ±

aκ1 κ2 + ω aκ1 κ2 + ω tan ⎡ (B1 x + B2 y − ρt + ϑ0) ⎤ ei (−κ1 x − κ2 y + ωt + ζ ), ⎥ ⎢ 2bκ1 2aB1B2 ⎦ ⎣

(26)

ψ (x , y, t ) = ±

aκ1 κ2 + ω aκ1 κ2 + ω cot ⎡ (B1 x + B2 y − ρt + ϑ0) ⎤ ei (−κ1 x − κ2 y + ωt + ζ ). ⎢ ⎥ 2bκ1 2aB1B2 ⎣ ⎦

(27)

The consequences (26) and (27) signify singular periodic solutions on condition that

aB1B2 (aκ1 κ2 + ω) > 0. 249

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Case-2: Eq. (11) can be yield by

aB1B2 (−(V ′)2 + 2VV ′ ′) − 4(ω + aκ1 κ2) V 2 − 8bκ1 V 3 = 0 through the medium of the transformation P =

1 V2.

(28)

Eq. (6) can be yield by

2

Q′ (ϑ) ⎞ Q′ (ϑ) ⎞ Q (ϑ) = δ0 + δ1 ⎛ + δ2 ⎛ ⎝ Q (ϑ) ⎠ ⎝ Q (ϑ) ⎠ ⎜

⎟

⎜

⎟

(29)

because of N = 2 which can be obtained by using of balancing rule The overdeterminet equations are acquired as follows Q−6 coeff.:

(Q′)2

′

or QQ′ with

Q3

in the ordinary differential equation (28).

8δ22 (Q′)6 (aB1B2 − bδ2 κ1) = 0,

(30)

Q−5 coeff.: 14δ2 (Q′) 4 ((aB1B2 δ1 − 2bδ1 δ2 κ1) Q′ − aB1B2 δ2 Q′ ′) = 0,

Q−4

(31)

coeff.:

(Q′)3 ((12aB1B2 δ0 δ2 + 3aB1B2 δ12 − 4aδ22κ1 κ2 − 24bδ0 δ22κ1 − 24bδ12δ2 κ1 − 4ωδ22) Q′ − 18aB1B2 δ1 δ2 Q′ ′ + 4aB1B2 δ22Q′ ′′) = 0,

Q−3

(32)

coeff.:

2(Q′)2 ((2aB1B2 δ0 δ1 − 4aδ1 δ2 κ1 κ2 − 24bδ0 δ1 δ2 κ1 − 4bδ13κ1 − 4ωδ1 δ2) Q′ + (−10aB1B2 δ0 δ2 − 2aB1B2 δ12) Q′ ′ + 3aB1B2 δ1 δ2 Q′ ′′) = 0, Q−2

(33)

coeff.:

− 6aB1B2 δ0 δ1 Q′Q′ ′ + (4aB1B2 δ0 δ2 + 2aB1B2 δ12) Q′Q′ ′′ + (4aB1B2 δ0 δ2 − aB1B2 δ12)(Q′ ′)2 + (−8aδ0 δ2 κ1 κ2 − 4aδ12κ1 κ2 − 24bδ0 2δ2 κ1 − 24bδ0 δ12κ1 − 8ωδ0 δ2 − 4ωδ12)(Q′)2 = 0,

(34)

Q−1 coeff.:

− 2δ0 δ1 ((4aκ1 κ2 + 12bδ0 κ1 + 4ω) Q′ − aB1B2 Q′ ′′) = 0, Q0

(35)

coeff.:

− 4aδ0 2κ1 κ2 − 8bδ03κ1 − 4ωδ0 2 = 0,

(36)

because of putting Eq. (29) in Eq. (28) and setting of the constant coefficients of the results are given by

4aB1 B2 (aκ1 κ2 + ω) , κ12 b2

δ0 = 0,

δ1 = ±

Q′ ′ = ±

4(aκ1 κ2 + ω) Q′, aB1B2

δ2 =

aB1B2 κ1 b

functions Q−l

to zero sequentially. The following

(37)

and

Q′ ′′ =

(38)

4(aκ1 κ2 + ω) Q′. aB1B2

(39)

The following equations can be reached as

Q′ = ±

aB1B2 ± k1 e 4(aκ1 κ2 + ω)

4(aκ1 κ2+ ω) ϑ aB1B2 ,

(40)

and

Q=

aB1B2 ± k1 e 4(aκ1 κ2 + ω)

4(aκ1 κ2+ ω) ϑ aB1B2

+ k2

(41)

with k1 and k2 integration constants by means of using Eqs. (38) and (39). The solitons of the Kundu–Mukherjee–Naskar model are emerged as follows 250

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⎧ ⎪ ψ (x , y, t ) = ± ⎨ ⎪ ⎩

4(aκ κ + ω)

4aB1 B2 (aκ1 κ2 + ω) κ12 b2

1 2 (B1 x + B2 y − ρt ) ⎞ ± aB1B2 ⎛± aB1B2 k1 e ⎟ ⎜ 4(aκ1κ2 + ω) 4(aκ κ + ω) ⎜ aB1B2 k e± aB11B22 (B1x + B2 y − ρt ) + k ⎟ 2 1 ⎠ ⎝ 4(aκ1κ2 + ω) 4(aκ κ + ω)

1 2 2

1 2 (B1 x + B2 y − ρt ) ⎞ ± aB1B2 ⎛± aB1B2 k1 e 4(aκ1 κ2 + ω) aB B ⎟ + κ1 b 2 ⎜ 4(aκ κ ω) 1 ⎜ aB1B2 k e± aB11B22+ (B1x + B2 y − ρt ) + k ⎟ 1 2 ⎠ ⎝ 4(aκ1κ2 + ω)

⎫ ⎪ i (−κ x − κ y + ωt + ζ ) e 1 2 ⎬ ⎪ ⎭

(42)

on account of inserting Eqs. (40) and (41) into Eq. (29). If we set

k1 =

4(aκ1 κ2 + ω) ± e aB1B2

4(aκ1 κ2+ ω) ϑ0 aB1B2 ,

k2 = ± 1,

(43)

we get: 1 2 aκ κ + ω aκ1 κ2 + ω ψ (x , y, t ) = ⎧− 1 2 sech2 ⎡ (B1 x + B2 y − ρt + ϑ0) ⎤ ⎫ ei (−κ1 x − κ2 y + ωt + ζ ). ⎥ ⎢ ⎬ ⎨ κ1 b aB1B2 ⎦⎭ ⎣ ⎩

(44)

The solution (44) points out bright soliton provided that

aB1B2 (aκ1 κ2 + ω) > 0. 1 2 aκ κ + ω aκ1 κ2 + ω ψ (x , y, t ) = ⎧ 1 2 csch2 ⎡ (B1 x + B2 y − ρt + ϑ0) ⎤ ⎫ ei (−κ1 x − κ2 y + ωt + ζ ). ⎢ ⎥⎬ ⎨ κ b aB B 1 1 2 ⎣ ⎦⎭ ⎩

(45)

The solution (45) points out singular soliton provided that

aB1B2 (aκ1 κ2 + ω) > 0. 1 2 aκ κ + ω 2 ⎡ aκ κ + ω ψ (x , y, t ) = ⎧− 1 2 sec (B1 x + B2 y − ρt + ϑ0) ⎤ ⎫ ei (−κ1 x − κ2 y + ωt + ζ ), − 1 2 ⎢ ⎥ ⎨ κ1 b aB1B2 ⎣ ⎦⎬ ⎩ ⎭

(46)

1 2 aκ κ + ω 2 ⎡ aκ κ + ω ψ (x , y, t ) = ⎧− 1 2 csc (B1 x + B2 y − ρt + ϑ0) ⎤ ⎫ ei (−κ1 x − κ2 y + ωt + ζ ). − 1 2 ⎢ ⎥ ⎨ κ1 b aB1B2 ⎣ ⎦⎬ ⎩ ⎭

(47)

The consequences (46) and (47) signify singular periodic solutions provided that

aB1B2 (aκ1 κ2 + ω) < 0. 3. Conclusions The Kundu–Mukherjee–Naskar model was examined for the sake of uncovering quite important optical soliton solutions. Dark, bright and singular solitons in addition to singular periodic solutions were yield with the modified simple equation technique along with parameter restrictions. The consequences acquired in this paper causes to consider the Kundu–Mukherjee–Naskar equation as more elaborate. For this reason, the model in DWDM technology is going to be considered with procedures which are trial equation procedure, extended Kudryashov's methodology, modified simple equation approach and lie symmetry analysis. The details of this very valuable work mentioned in this section is going to be presented respectively. References [1] M. Ekici, A. Sonmezoglu, A. Biswas, M.R. Belic, Optical solitons in (2 + 1)-dimensions with Kundu–Mukherjee–Naskar equation by extended trial function scheme, Chin. J. Phys. 57 (2018) 72–77. [2] Y. Yıldırım, Optical solitons to Kundu–Mukherjee–Naskar model with trial equation approach, Optik (2019) (in press). [3] E. Yaşar, Y. Yıldırım, A.R. Adem, Perturbed optical solitons with spatio-temporal dispersion in (2 + 1)-dimensions by extended Kudryashov method, Optik 158 (2018) 1–14. [4] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation for complex Ginzburg–Landau equation with modified simple equation method, Optik 158 (2018) 399–415. [5] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical solitons for Lakshmanan–Porsezian–Daniel model by modified simple equation method, Optik 160 (2018) 24–32. [6] A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with resonant nonlinear Schrödinger's equation having full nonlinearity by modified simple equation method, Optik 160 (2018) 33–43. [7] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with full nonlinearity for Kundu–Eckhaus equation by modified simple equation method, Optik 157 (2018) 1376–1380. [8] A. Biswas, Y. Yildirim, E. Yasar, H. Triki, A.S. Alshomrani, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with Gerdjikov–Ivanov equation by modified simple equation method, Optik 157 (2018) 1235–1240.

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