mustafa inc, salathiel yakada, depelair bienvenu, gambo

Research Article

Mustafa Inc, Salathiel Yakada, Depelair Bienvenu, Gambo Betchewe, and Yu-Ming Chu*

W-Chirped optical solitons and modulationinstability analysis of Chen–Lee–Liu equation inoptical monomode fibres

https://doi.org/10.1515/phys-2021-0003received October 13, 2020; accepted January 19, 2021

Abstract: This work was devoted to unearth W-chirped tothe famous Chen–Lee–Liu equation (CLLE) in opticalmonomode fibres. The results obtained will be useful toexplain wave propagating with the chirp component. Toattempt the main goal, we have used the new sub-ordinary differential equation (ODE) technique whichwas upgraded recently by Zayed EME, Mohamed EMA.Application of newly proposed sub-ODE method to locatechirped optical solutions to the Triki–Biswas equationequation. Optik. 2020;207:164360. On the other hand,we have used the modulation analysis to study the steadystate of the obtained chirped soliton solutions in opticalmonomode fibres.

Keywords: W-chirped solitons, Chen–Lee–Liu equation,modulation instability

1 Introduction

Today, many authors focused their interest on treatmentof the nonlinear physical system to unearth the wavecalled “soliton.” Solitons have been found in many

different physical systems such as bright soliton, darksoliton, breather-like solitons, and abundant vector soli-tons [1–14]. The prototypical model for the study of thesesolitons is the nonlinear Schrödinger (NLS) equation withattractive or repulsive interactions [15]. Many experi-ments suggested that the NLS model described well theevolution dynamics of many different nonlinear systems,such as water wave tank, optical fibre, and Bose–Einsteincondensate [16]. The propagation of chirped soliton inthe nonlinear physical system has become nowadays awide field of research. During the spread of the latter, thepulse is loosed in nonlinear physical system because ofmany kinds of perturbations and properties of materials[17]. Today, with the event of chirped soliton which isable to remain in its form after perturbation or collision,it is possible to make pulse uniform in the nonlinearphysical system [18]. By exciting chirped soliton, it ispossible to amplify or reduce a pulse which propagatesin the nonlinear physical system [19]. Otherwise, theW-chirped soliton as mentioned in the title is a chirpedsoliton with W-shape. In short, the solitons are full ofmany interests in various fields of applications such ascommunication, medicine, hydrodynamic just to name afew [20].

In addition, other models have been designed inrecent years to illustrate the propagation of optical waves,among which the the Triki–Biswas equation [21], Fokas–Lenells equation [22] and so on. In this logic, the CLLEmodel was built to describe soliton in monomode opticalfibres. It is given in the following form [23]:

iq aq ib q q 0,t xx x2

+ + ∣ ∣ = (1)

where the quantity q x t,( ) is the wave profile in the spatialand temporal representation. Regarding the parameters aand b they are group velocity dispersion (GVD) and thecontribution of the self-phase modulation (SPM), respec-tively [23]. More recently, the model has been the subjectof a treatment taking into account the dual power low ofnonlinearity, as a result the chirped solitons have beenobtained [23]. Our objective in this article is to construct

Mustafa Inc: Department of Mathematics, Science Faculty, FiratUniversity, Elazig, 23119, Turkey; Department of Medical Research,China Medical University Hospital, China Medical University,Taichung, TaiwanSalathiel Yakada: Faculty of Industrial Engineering of the Universityof Douala, Douala, CameroonDepelair Bienvenu, Gambo Betchewe: Department of Physics,Faculty of Science, The University of Maroua, P.O. Box 814, Maroua,Cameroon

* Corresponding author: Yu-Ming Chu, Department of Mathematics,Huzhou University, Huzhou, 313000, China; Hunan Provincial KeyLaboratory of Mathematical Modeling and Analysis in Engineering,Changsha University of Science and Technology, Changsha, 410114,China, e-mail: [email protected]

Open Physics 2021; 19: 26–34

Open Access. © 2021 Mustafa Inc et al., published by DeGruyter. This work is licensed under the Creative Commons Attribution 4.0International License.

https://doi.org/10.1515/phys-2021-0003

mailto:[email protected]

new forms of chirped solitons using the improved sub-ordinary differential equation (ODE) method by Zayedand Mohamed [21]. To achieve this, we will use the fol-lowing diagram: Section 1 uses the usual transformationhypothesis to obtain the ordinary differential equation,while Section 2 is devoted to the application of the sub-ODE method. In Section 3, graphical representations intwo-dimensional (2D) and three-dimensional (3D) aremade to illustrate the physical explanation of the obtainedresults and it will be followed by the conclusion of thework in Section 4.

2 Mathematical investigation andchirped soliton to CLLE

To turn to the ODE, the following transformation isassumed:

q x t ϕ ξ e, ,i f ξ tΩ( ) = ( )

( ( )− ) (2)

where ξ κx vt= − . The parameters ϕ ξ( ) and f ξ( ) arereals, while the chirp component is written as follows:

δω x tx

f ξ t κf ξ, Ω .( ) = −

∂

∂

[ ( ) − ] = − ′( ) (3)

Using equations (2) and (1), it leads to two nonlinearordinary equations, where the first one is the imaginarypart and the second one is real.

vϕ aκ f ϕ aκ f ϕ bκϕ ϕ2 02 2 2− ′ + ′ ′ + ″ + ′ = (4)

and

vf ϕ ϕ aκ ϕ aκ f ϕ bκf ϕΩ 0.2 2 2 3′ + + ″ − ′ − ′ = (5)

To obtain the analytical form of chirps, we multiply equa-tion (4) by ϕ, and then integrate once by simultaneouslyconsidering the integration constant as zero as follows:

v ϕ aκ ϕ f bκ ϕ2 4

0.2 2 2 4− + ′ + = (6)

From equation (6), it is obtained that

f vaκ

baκ

ϕ2 4

,22

′ = − (7)

and this allows us to get the expression of the chirp asfollows:

δω x t vaκ

ba

ϕ,2 4

.2( ) = − + (8)

Introducing equation (7) into equation (5) leads to thefollowing nonlinear ordinary equation:

ϕ ϕ ϕ ϕ 0,0 13

2 35

ℓ + ℓ + ℓ ″ + ℓ = (9)

where

v aκaκ

bvaκ

aκ ba

14

4Ω , 12

,

, 316

.

02 2

2 1

22

32

ℓ =

+

ℓ = −

ℓ = ℓ =

(10)

To arrive at an integrable form of equation (9), weconjecture ϕ H2

= , which makes it possible to obtain thefollowing form:

H H H H H H4 4 2 4 0.02

13

22

34

ℓ + ℓ + ℓ [ ″ − ′ ] + ℓ = (11)At the moment, we assume the solutions of equation (11)as in [21],

H μF ξ ,n= ( ) (12)

with m a parameter and F ξ( ) is expressed as follows seeref. [21],

F ξ AF ξ BF ξ CF ξ DF ξ

EF ξ p, 0.

p p p

p

2 2 2 2 2 2

2 2

′ ( ) = ( ) + ( ) + ( ) + ( )

+ ( ) >

− − +

+

(13)

We will now use the balance principle betweenthe terms H H″ and H4, which leads to writen n p n p n2 4+ + = ⇒ = .

From there we get the solutions of equation (11),which is written as follows:

H μF ξ .p= ( ) (14)

Making use of equation (12) with equation (13) in equa-tion (11), the system of equations depending on F ξjp

( ),j 0, 2, 3, 4= ( ) is built as follows:

l μ l μ p C F ξ l μ p D l μ F ξ

l μ p E l μ F ξ l μ p A

4 2 4

3 4 0.

p p

p

02

22 2 2

22 2

13 3

22 2

34 4

22 2

( + ) ( ) + ( ( ) + ) ( )

+ ( + ) ( ) − =

(15)

Obtaining the constants defined in equation (13) is depen-dent on the resolution of equation (15) by making use ofthe mathematical tool MAPLE.

Next, we employ equation (10) to point out the fol-lowing result:

A 0= , B B= , C v aκa κ p

4Ω2 2

2 4 2= −

+

, D vbμa κ p2 3 2= ,

E b μa κ p

14

2 2

2 2 2= − .

Case 1. A 0= , and assume that B 0= , the following caseis obtained

According to the new type of algorithm set out in ref.[21], the following solutions should be revealed• i.1 v a κ4 Ω 0,2 2

+ < D C2< , and μ =

a v aκ pb

12

Ω 4ΩΩ

2 2±

( + )

, what follows from E C.DC4

2= −

W-chirped optical solitons and MI analysis in OMF 27

q x t μ

C ξ DC

e,cosh

2

,i f ξ t1

Ω

12

( ) =

( ) −

×( ( )− ) (16)

δω x t vaκ

bμa C ξ D

C

,2 4

1

cosh2

.1( ) = − +

( ) −

(17)

• i.2 v a κ E4 Ω 0, 0.2 2+ = <

q x t aκ

bbκD D

DξE e, 2 4 4 ,i f ξ t

2 2Ω

12

( ) =

( )

− ×( ( )− ) (18)

thus the chirp solutions are obtained

δω x t v

aκbμ

aD

DξE,

2 44 4 .2 2( ) = − +

( )

− (19)

Case 2. Considering A B 0= = , v a κ4 Ω 02 2+ < , we have

gained combined bright soliton and hyperbolic functionsas solutions• ii.1 D CE4 02

− > ,

q x tμC C ξ

D CE D CE D C ξ

e

,4 sech 1

2

2 4 4 sech 12

,

,i f ξ t

3

2

2 2 2

12

Ω

( ) =

− − ( − + )

×( ( )− ) (20)

q x tμC C ξ

D CE D CE D C ξ

e

,4 csch 1

2

2 4 4 csch 12

,

,i f ξ t

4

2

2 2 2

12

Ω

( ) =

− + ( − − )

×( ( )− ) (21)

q x t μCD CE C ξ D

e, 24 cosh

,i f ξ t5 2

12

Ω( ) =

± − ( ) −

×( ( )− ) (22)

chirp solutions correspond to

δω x t vaκ

ba

C C ξ

D CE D CE D C ξ

,2

4

4 sech

2 4 4 sech,

3

2 12

2 2 2 12

( )

( )

( ) = −

+

− − ( − + )

(23)

δω x t vaκ

bμa

C C ξ

D CE D CE D C ξ

,2

4

4 csch

2 4 4 csch,

4

2 12

2 2 2 12

( )

( )

( ) = −

+

− + ( − − )

(24)

δω x t vaκbμ

aC

D CE C ξ D

,2

42

4 cosh.

5

2

( ) = −

+

± − ( ) −

(25)

Figure 1 plots (h1) the corresponding W-chirp brightsoliton of equation (17) and the bright soliton equation(16) when the speed of the soliton is v a κ4 Ω 0,2 2

+ <

and the parameter of the sub-ODE method is given by

μa v aκ p

b12

Ω 4ΩΩ

2 2= ±

( + )

. Furthermore, Figure 2 is the

spatiotemporal plot 2D of equations (16) and (17), respec-tively, at the same constraint relation on the velocity andfree parameter of the sub-ODE algorithm. Moreover,Figure 3 is the spatiotemporal plot of the dark solitonsolutions of analytical solution equation (25).

(h1)

t

x

40

200-100 -50 0 50 0100

0.5

× 10-5

|δω1(x,t)|

1

(h2)

t

x

40

200

-100-50

050 0100

0.5

|q1(x,t)|2

1

Figure 1: Plot of the corresponding h1( ) W-chirp bright soliton and (h2) the bright soliton of equation (16) at a 1= , μ 1= , κ 3.15= ,b 0.2003= , Ω −0.304215= , v 0.12= , C 0.00307= .

28 Mustafa Inc et al.

• ii.2 D CE4 0,2− <

q x t μCD CE C ξ D

e

, 24 sinh

,i f ξ t

6 2

12

Ω

( ) =

± −( − ) ( ) −

×( ( )− )

(26)

and the chirp

δω x t vaκbμ

aC

D CE C ξ D

,2

42

4 sinh.

6

2

( ) = −

+

± −( − ) ( ) −

(27)

• ii.3 D CE4 0,2− =

q x t μC

Dp C ξ e, 1 tanh2

,i f ξ t7

12 Ω

( ) = − ± ×( ( )− )

(28)

q x t μCD

C ξ e, 1 coth2

,i f ξ t8

12

Ω( ) = − ± ×

( ( )− ) (29)

chirps are expressed as

δω x t vaκ

bμa

CD

C ξ,2 4

1 tanh2

,7( ) = − + − ± (30)

δω x t vaκ

bμa

CD

C ξ,2 4

1 coth2

,8( ) = − + − ± (31)

and two other solutions when the constraint condition islimited to A B v a κ0, 4 Ω 02 2

= = + < ,

q x tμCD C ξ

D CE C ξ

e

,sech

2

1 tanh2

,i f ξ t

9

2

22

12

Ω

( ) =

− ±

×( ( )− )

(32)

q x tμCD C ξ

D CE C ξ

e

,csch

2

1 coth2

,i f ξ t

10

2

22

12

Ω

( ) =

− ±

×( ( )− )

(33)

and the corresponding chirp

-50 0 50x

0

0.2

0.4

0.6

0.8

1

| q1(x,t=0)|2

(h4)

-50 0 50x

0

1

2

3

4

|δω1(x,t=0)|

× 10-3(h

3)

Figure 2: Plot of the corresponding h3( ) W-chirp bright soliton 2-D and (h4) the bright soliton 2-D of equation (16) at a 1= , μ 1= , κ 3.15= ,b 0.2003= , Ω −0.304215= , v 0.12= , C 0.00307= .

x

t

40

30

200

-100 10-500

50 0100

5

|δω5(x,t)|

10

Figure 3: Spatiotemporal plot of equation (25) at a 1= , μ 0.0001= ,κ 1.47= , b 0.0023= , Ω −8.89= , v −2.75= , D −1.72= ,E −4.63= , C 3.70= .


δω x t vaκ

bμa

CD ξ

D CE C ξ

,2

4

sech

1 tanh2

,C

9

22

22

( )

( ) = −

+

− ±

(34)

δω x t vaκ

bμa

CD ξ

D CE C ξ

,2

4

csch

1 coth2

.C

10

22

22

( )

( ) = −

+

− ±

(35)

Case 3. Considering A B 0= = , and v a κ4 Ω 02 2+ > , the

following bright soliton and hyperbolic functions as solu-tions is obtained• iii.1 D CE4 0,2

− >

q x t

Cμ C ξ

D CE D CE D C ξ

e

,

2 sec2

2 4 4 sec2

,

,i f ξ t

11

2

2 2 2

12

Ω

( )

=

−

−

− − ( − − )

−

×( ( )− ) (36)

q x t

Cμ C ξ

D CE D CE D C ξ

e

,

2 csc2

2 4 4 csc2

,

,i f ξ t

12

2

2 2 2

12

Ω

( )

=

−

− + ( − + )

−

×( ( )− ) (37)

q x t Cμ C ξD CE D C ξ

e

, 2 sec4 sec

,i f ξ t

13 2

12

Ω

( ) =

( − )

± − − ( − )

×( ( )− )

(38)

q x t μC C ξD CE D p C ξ

e

, 2 csc4 csc

,i f ξ t

14 2

12

Ω

( ) =

( − )

± − − ( − )

×( ( )− )

(39)

the corresponding chirp-like solution

δω x t vaκ

bμa

C ξ

D CE D CE D ξ

,2

4

2 sec

2 4 4 sec,

C

C

11

22

2 2 22

( )

( )

( ) = −

+

−

− − ( − − )

−

−

(40)

δω x t vaκ

bμa

C ξ

D CE D CE D ξ

,2

4

2 csc

2 4 4 csc,

C

C

12

22

2 2 22

( )

( )

( ) = −

+

− + ( − + )

−

−

(41)

δω x t vaκbμ

aC C ξ

D CE D C ξ

,2

42 sec

4 sec,

13

2

( ) = −

+

( − )

± − − ( − )

(42)

δω x t vaκbμ

aC C ξ

D CE D C ξ

,2

42 csc

4 csc.

14

2

( ) = −

+

( − )

± − − ( − )

(43)

Case 4. For A 0= , B CD

827

2= , E D

C4

2= , we obtain the fol-

lowing cases• iv.1 v a κ4 Ω 0,2 2

+ > we gained hyperbolic functionsolutions

q x tμC C ξ

D C ξ

e

,8 tanh 1

2 3

3 3 tanh 12 3

,i f ξ t

15

2

2

12

Ω

( ) = −

−

+

−

×( ( )− )

(44)

q x tμC C ξ

D C ξ

e

,8 coth 1

2 3

3 3 coth 12 3

,i f ξ t

16

2

2

12

Ω

( ) = −

−

+

−

×( ( )− )

(45)

the chirp hyperbolic function solutions

δω x t vaκ

bμa

C ξ

D ξ

,2

4

8 tanh

3 3 tanh,

C

C

15

2 12 3

2 12 3

( )

( )

( ) = −

+ −

+

−

−

(46)


δω x t vaκ

bμa

C ξ

D ξ

,2

4

8 coth

3 3 coth,

C

C

16

2 12 3

2 12 3

( )

( )

( ) = −

+ −

+

−

−

(47)

• iv.2 v a κ4 Ω 0,2 2+ < we gained trigonometric function

solutions

q x tμC C ξ

D C ξe,

8 tan 12 3

3 3 tan 12 3

,i f ξ t17

2

2

12

Ω( ) =

−

×( ( )− ) (48)

q x tμC C ξ

D C ξe,

8 cot 12 3

3 3 cot 12 3

,i f ξ t18

2

2

12

Ω( ) =

−

×( ( )− ) (49)

the equivalent chirp is obtained

δω x t vaκ

bμa

μC ξ

D ξ,

2 4

8 tan

3 3 tan,

C

C17

2 12 3

2 12 3

( )

( )

( ) = − +

−

(50)

δω x t vaκ

bμa

C ξ

D ξ,

2 4

8 cot

3 3 cot.

C

C18

2 12 3

2 12 3

( )

( )

( ) = − +

−

(51)

Case 5. For A B 0= = ,• v.1 v a κ4 Ω 0,2 2

+ <

q x t μCee D CE

e, 44

,C ξ

C ξi f ξ t

19

12

Ω( ) =

( − ) −

×

( )

±

( ( )− ) (52)

δω x t v

aκbμ

aCe

e D CE,

2 44

4.

p C ξ

C ξ19( ) = − +

( − ) −

( )

±

(53)

Case 6. For A 0= , in what follows, Jacobian elliptic func-tion solutions will be presented. It should be noted thatthese solutions can turn to soliton solutions such asbright, dark solitons and periodic or singular functionswhen m 1→ or m 0.→ Here the parameter m is theJacobian modulus.

Having already obtained bright and dark solitonsolutions, periodic and singular solutions above, we willlimit ourselves strictly to the Jacobian elliptic solutionstype. They look like

• vi.1 For E 0< , B Dm E32

3

2 2= , C D mm E

4 116

2 2

2=

( + )

, it is

revealed

q x t μDE

mcn Dm E

ξ

e

,4

14

1

,i f ξ t

20

12

Ω

( ) = − ± −

×( ( )− )

(54)

q x t μDE

m sn Dm E

ξ

dn Dm E

ξ

e

,4

11

41

41

,i f ξ t

21

2

12

Ω

( ) = − ±

− −

−

×( ( )− )

(55)

δω x t vaκbμ

aDE

mcn Dm E

ξ

,2

4 41

41 ,

20( ) = −

+ − ± −

(56)

δω x t vaκ

bμa

DE

m sn Dm E

ξ

dn Dm E

ξ

,2 4

41

14

1

41

.

21

2

12

( ) = − +

× − ±

− −

−

(57)

• vi.2 For E 0< , B m Dm E32 1

2 3

2 2=

( − )

,C D mm E5 41

16 1

2 2

2=

( − )

( − )

, it is

revealed

q x t

μDE m

dn Dm E

ξ

e

,

41 1

1 41

1,

,i f ξ t

22

2 2

12

Ω

( )

= − ±

−

−

( − )

×( ( )− )

(58)

q x t μDE

dn Dm E

ξ

e

,4

1 1

41

1

,i f ξ t

23

2

12

Ω

( ) = − ±

−

( − )

×( ( )− )

(59)


then the chirps

δω x t vaκbμ

aDE m

dn

Dm E

ξ

,2

4 41 1

1

41

1,

22

2

2

( ) = −

+ − ±

−

× −

( − )

(60)

δω x t vaκ

bμa

DE dn ξ

,2

4 41 1 .

Dm E

23

41

1 2( )

( ) = −

+ − ±

−( − )

(61)

• vi.3 For E 0< , B m DE32

2 3

2= , C D mE

4116

2 2=

( + )

, hence

q x t aκb

bκD DE

dn DE

ξ

e

, 24

14

1 ,

,i f ξ t

24

12

Ω

( ) = − ± −

×( ( )− )

(62)

q x t aκb

bκD DE

m

dn DE

ξ

e

, 24

1 1

41

,

,i f ξ t

252

12

Ω

( ) = − ±

−

−

×( ( )− )

(63)

which leads to chirp solutions as follows:

δω x t vaκ

bμa

DE

dn DE

ξ,2 4 4

14

1 ,24( ) = − + − ± − (64)

δω x t vaκ

bμa

DE

m

dn DE

ξ

,2

4 41 1

41

.

25

2

( ) = −

+ − ±

−

−

(65)

3 Modulation instability (MI)analysis

After obtaining abundant analytical solutions, it is judi-cious to seek the stability of these solutions by using the

famous linear stability technique in equation (1). To getthere, we project the perturbed solution of equation (1) inthe following form:

q x t P A x t e ϕ cP x, , , .iϕ0 NL 0NL( ) = [ + ( )] = (66)

With the incident power P0, A x t,( ) is the so-called implicitperturbation function, while the nonlinear phase is repre-sented by ϕNL. Introduce equation (70) into equation (1),then linearizing with respect to A x t,( ), it stems

iA aA ibP A 0.t xx x0+ + = (67)

Let us estimate the form of the solution of equation (71) asfollows:

A x t a e a e, ,i Kx t i Kx t1

Ω2

Ω( ) = +

( − ) − ( − ) (68)

the parameters K and Ω represent the wave numberand the perturbation frequency, respectively. By

0-20

-15

-10

-5

0

5

10

15

20

b=0.5b=0.3b=0.2

-200

0.5

1

1.5

2

2.5

Ω[Gain]/m

Ω[Gain]/m

b=0.2b=0.3b=0.4

1 2 3 4 5 6

0 20 40 60 80 100

G(Ω)

G(Ω)

Figure 4: MI gain spectrum G mΩ −1( ) [ ] as a function of frequency

Ω (Hz) in normal GVD a 1( = ) with (d1) the positive effect of SPM and(d2) the negative effect of SPM at the power incident P 1,5000 = .


inserting equation (72) into equation (71), a system ofequations in terms of a1 and a2 is obtained after separ-ating the coefficients of ei Kx tΩ( − ) and e i Kx tΩ− ( − ).

aK bP K aΩ 0,20 1( − − ) = (69)

aK bP K aΩ 0,20 2(− − + ) = (70)

then the dispersion relation is as follows:

bP K a K b P KΩ 2Ω 0.20

2 4 202 2

− + + − = (71)

The solution of equation (75) can be obtained as follows:

Ka

P b P b a a bP12

2 2 4 Ω 8 Ω ,02 2

04 4 2 2 2

0= ± − − − (72)

The obtained solution depends on two essential termsof the monomode fibre, namely the GVD and the SPM.Furthermore, the obtained solution equation (76) makesit possible to locate the stability of the stationary state.So, for

P b P b a a bP2 2 4 Ω 8 Ω 0,02 2

04 4 2 2 2

0− − − ≥ (73)

or

P b P b a a bP2 2 4 Ω 8 Ω ,02 2

04 4 2 2 2

0≥ − − (74)

this solution is purely real, and there is a saturation of thestationary state in the face of small perturbations. On theother hand, if

P b P b a a bP2 2 4 Ω 8 Ω ,02 2

04 4 2 2 2

0< − − (75)

the wave number K has an imaginary part, and this solu-tion is unstable in the steady state because of the expo-nential growth of the perturbation. There is no doubt thatthe MI takes place. From there, we can get the MI gainspectrum as follows:

G aIm

P b P b a a bP

Ω Ω

2 2 4 Ω 8 Ω .02 2

04 4 2 2 2

0

( ) = ( )

= − − −

(76)

Figure 4 depicts the MI gain spectrum with the effectof SPM, which is related to the kerr nonlinearity of themonomode optical fibres. Inspecting the curve, tworegimes are setting out. In the first case, we assume thepositive SPM and the second one with negative valuedof SPM.

4 Conclusion

This article concerns the study of the W-chirped solitonand other solutions in optical monomode fibres by using

the dimensionless CLLE. More recently, chirped soliton tothe model with dual power law of nonlinearity has beenstudied. The steady state of the obtained results has notbeen made. In view of the importance of the chirpedsignal in the communication system, we undertook themodulation analysis of the steady state of the obtainedresults in order to be able to define their domain of exis-tence through graphical representations of the gain spec-trum. The perturbation effect to will be added next themodel to better circumscribe the constraints linked to thepropagation of the chirped signal with perturbationeffects.

Declaration of competing interests: The authors declarethat they have no known competing financial interests orpersonal relationships that could have appeared to influ-ence the work reported in this article.

Funding: This work was supported by the Natural ScienceFoundation of China (Grant Nos. 61673169, 11301127,11701176, 11626101, and 11601485).

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mustafa inc, salathiel yakada, depelair bienvenu, gambo

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