Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber

Khalid K. Ali; Tasneem Adel; Mansour N. Abd El-Salam; Mohamed S. Mohamed; Mohamed A. Shaalan

doi:10.1515/phys-2025-0164

Artikel Open Access

Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber

Khalid K. Ali , Tasneem Adel , Mansour N. Abd El-Salam , Mohamed S. Mohamed und Mohamed A. Shaalan

Veröffentlicht/Copyright: 5. Juni 2025

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Abstract

In this study, we use the powerful and strong method to obtain a solution of the Sasa–Satsuma equation as ( ℋ + G ′ G 2 ) -expansion method. This method plays a considerable role in solving nonlinear partial differential equations (NPDEs). We investigate the modulation instability in higher-order NPDEs. Modulation instability is a phenomenon observed in certain types of nonlinear systems, such as optical fiber or plasma waves. Modulation instability is a key process in generating optical solitons, rogue waves, and interest in various fields such as nonlinear optics and plasma physics. Using a linearizing technique, we establish the modulation instability and show the influence of a higher nonlinear component in modulation instability. We examine the bifurcation analysis of the Sasa–Satsuma equation. The time histories and Poincare mapping are used to scrutinize the chaotic behaviors of the dynamical system of the Sasa–Satsuma equation excited by a parametric excitation force. To control the vibrating system, use proportional feedback control (P-Controller). Two-dimensional and three-dimensional figures are presented for singular, dark, and bright optical soliton solutions related to optical fiber. These graphs are very important and useful in describing the behavior of solutions.

Keywords: Sasa–Satsuma equation; (ℋ + G′/G2)-expansion method; nonlinear optical system; qualitative analysis; dynamical system; P-controller

1 Introduction

Nonlinear partial differential equations (NPDEs) are a type of mathematical equation. NPDEs are more complex and challenging to solve. They arise in various fields, including physics, engineering, and finance, to model diverse phenomena like fluid dynamics, wave propagation, and phase transitions. Solving NPDEs often requires advanced techniques such as numerical methods, perturbation methods, or transforming them into equivalent integral equations. The solutions of NPDEs can lead to fascinating and intricate patterns, making them an essential tool in understanding many real-world applications [1–9].

The soliton solutions of NPDEs such as Schrödinger equation [2], Akshmanan–Prosezain–Daniel equation [10], Navier–Stokes equation [11], Korteweg–De Vries equation [12], Burgers’ equation [13], higher-order nonlinear Schrödinger equation [14], Biswas–Arshed equation [15], Kundu–Eckhaus equation [16], and Davey–Stewartson–Kadomtsev–Petviashvili equation [17] are studied by different techniques.

There are several methods to solve the Sasa–Satsuma equation, such as Darboux transformation [18], Sinh–Gordon expansion method [19], modified Kudryashov method [20], F-expansion method [21], modified simple equation method [22], improved modified extended tanh scheme [23], new version of Kudryashov and exponential method [24], trial equation method [25], ∂ -dressing method[26], ∂ method [27], extend trial equation method, and generalized Kudryashov method [28].

In this work, we solve the Sasa–Satsuma equation [29,30]

(1) a ψ x x + b ψ ∣ ψ ∣ 2 + i ( γ ψ ( ∣ ψ ∣ 2 ) x + β ψ x ∣ ψ ∣ 2 + α ψ x x x ) + i ψ t = 0 ,

where ψ ( x , t ) is the complex valued wave function, x , t are space and time variables, respectively, and b , γ , β , and α are real parameters. The Sasa–Satsuma equation is a higher-order nonlinear Schrödinger equation. It is significant in the field of optics as it describes the propagation of femtosecond pulses in optical fibers.

We solve the previous equation by using the ( ℋ + G ′ G 2 ) -expansion method [31–33]. We also discuss the modulation instability and the influence of nonlinearity in modulation instability. Modulation instability refers to a phenomenon that occurs in nonlinear optical systems, particularly in optical fiber. It is a process where small perturbations in the intensity of an optical signal can grow rapidly as it propagates through the fiber [34–36]. The bifurcation and chaotic behaviors of traveling wave solutions are discussed in dynamical systems [37–41]. Atepor and Akoto [42] studied the effect of parametric force on an auto-parametric vibration absorber system. Amer et al. [43] investigated the vibration analysis and dynamic responses of a hybrid Rayleigh–Van der Pol–Duffing oscillator using a proportional-derivative controller. The average method was employed to obtain the approximate solution of the vibrating system.

This work is organized as follows: Sections 2 and 3 present the description and strategy of the ( ℋ + G ′ G 2 ) -expansion method. Section 4 describes the applications of the proposed methods to find solutions to nonlinear models. Section 5 illustrates the graphical representations in 2D and 3D to show the behavior of solutions. Section 6 discusses modulation instability analysis (MIA). Section 7 exhibits bifurcation analysis. Section 8 displays the chaotic analysis. Finally, Section 9 presents the conclusion of our work.

2 Description of the method

In this section, we introduce the steps of the proposed method to show how to apply this method on Sasa–Satsuma Eq. (1). Consider the partial differential equation.

(2) D = ( ψ , ψ t , ψ x , ψ t t , ψ x x , … ) ,

where D represents a polynomial comprising the unknown function ψ = ψ ( x , t ) as well as its different partial derivatives.

By using the transformation

(3) φ ( x , t ) = ζ + t ω + ( − κ ) x , ξ = η ( x − ν t ) , ψ ( x , t ) = U ( ξ ) exp ( i φ ( x , t ) ) ,

where ω , κ , ν , ζ , η are arbitrary constants. Substitute (3) in (2), then (2) becomes ordinary differential equation as follows:

(4) S = ( U ′ , U ″ , U ‴ , … ) ,

where S is a polynomial in U ( ξ ) and its derivatives.

3 Strategy of the ( ℋ + G ′ G 2 ) -expansion approach

The essential steps of the ( ℋ + G ′ G 2 ) -expansion technique are as follows:

Presume the solution of (4) is given by:
(5) U ( ξ ) = ∑ i = − N N a i F ( ξ ) i ,
where F ( ξ ) = ( ℋ + G ′ ( ξ ) G 2 ( ξ ) ) , G ′ G 2 satisfies the differential equation of second order:
(6) G ′ G 2 ′ = A + B G ′ G 2 2 + C G ′ G 2 ,
where a i are undetermined constants, A , B , and C are constants to be chosen later.
Utilizing the homogeneous balance principle to determine the positive integer N .
We procure the following classes of solutions of (6):
Class 1: When A B > 0 , C = 0 :
(7) G ′ G 2 = A B P cos ( A B ξ ) + Q sin ( A B ξ ) Q cos ( A B ξ ) − P sin ( A B ξ ) ,
where P and Q are arbitrary constants not equal to zero.
Class 2: When A B < 0 , C = 0 :
(8) G ′ G 2 = − ∣ A B ∣ B P sinh ( 2 ∣ A B ∣ ξ ) + P cosh ( 2 ∣ A B ∣ ξ ) + Q P sinh ( 2 ∣ A B ∣ ξ ) + P cosh ( 2 ∣ A B ∣ ξ ) − Q .
Class 3: When A = 0 , B ≠ 0 , C = 0 :
(9) G ′ G 2 = − P B ( P ξ + Q ) .
Class 4: When C 2 − 4 A B ≥ 0 , C ≠ 0 :
(10) G ′ G 2 = − C 2 − 4 A B P cosh 1 2 ξ C 2 − 4 A B + Q sinh 1 2 ξ C 2 − 4 A B ( 2 B ) P sinh 1 2 ξ C 2 − 4 A B + Q cosh 1 2 ξ C 2 − 4 A B − C 2 B .
Class 5: When C 2 − 4 A B ≤ 0 , C ≠ 0 :
(11) G ′ G 2 = − − C 2 + 4 A B P cos 1 2 ξ − C 2 + 4 A B − Q sin 1 2 ξ − C 2 + 4 A B ( 2 B ) P sin 1 2 ξ − C 2 + 4 A B + Q cos 1 2 ξ − C 2 + 4 A B − C 2 B .
Insert (5) and (6) in (4); a system of equations is emerged as the coefficients with the same powers of ( ℋ + G ′ G 2 ) disappear. The Mathematica program is used to obtain the solution of this system.

4 Applications of methods

By inserting (3) into (1) and equating the real and imaginary parts to zero, we obtain

(12) η ( η ( ( 3 α κ + a ) U ″ ( ξ ) + i α η U ( 3 ) ( ξ ) ) − i U ′ ( ξ ) ( 3 α κ 2 + 2 a κ + ν ) ) − U ( ξ ) ( κ 2 ( α κ + a ) + ω ) + U ( ξ ) 3 ( β κ + b ) + i η ( β + 2 γ ) U ( ξ ) 2 U ′ ( ξ ) = 0 .

A real part is obtained as

(13) η 2 ( 3 α κ + a ) U ″ ( ξ ) − U ( ξ ) ( α κ 3 + a κ 2 + ω ) + U ( ξ ) 3 ( β κ + b ) = 0 ,

and imaginary part is given by

(14) − U ( ξ ) ( 3 α κ 2 + 2 a κ + ν ) + α η 2 U ″ ( ξ ) + 1 3 ( β + 2 γ ) U ( ξ ) 3 = 0 .

Equating the coefficients of U ( ξ ) , U ( ξ ) 3 , and U ″ ( ξ ) in Eqs (13) and (14), respectively, we obtain the following:

(15) ω = ν ( 3 α κ + a ) + 2 κ ( 2 α κ + a ) 2 α , b = a ( β + 2 γ ) 3 α + 2 γ κ .

We balance U ( ξ ) 3 , and U ″ ( ξ ) in Eq. (13), then, we obtain N = 1.

4.1 Solution of ( ℋ + G ′ G 2 ) -expansion approach

Substitute (5) and (6) in (13) with N = 1 , then collect all terms that have the same power of ( ℋ + G ′ G 2 ) and equate them to zero, we obtain the following system:

− 2 a 2 a 0 κ α + a a 0 3 β 3 α + 2 a a − 1 a 0 a 1 β α + 2 a a 0 3 γ 3 α + 4 a a − 1 a 0 a 1 γ α − 9 α a 0 κ 3 − a a 0 ν α − 6 α a 1 A B η 2 κ ℋ − 2 a a 1 A B η 2 ℋ + 3 α a 1 A C η 2 κ + a a 1 A C η 2 + a 0 3 β κ + 6 a − 1 a 0 a 1 β κ − 6 α a 1 B 2 η 2 κ ℋ 3 − 2 a a 1 B 2 η 2 ℋ 3 − 6 α a − 1 B 2 η 2 κ ℋ − 2 a a − 1 B 2 η 2 ℋ + 3 α a − 1 B C η 2 κ + a a − 1 B C η 2 + 9 α a 1 B C η 2 κ ℋ 2 + 3 a a 1 B C η 2 ℋ 2 + 2 a 0 3 γ κ + 12 a − 1 a 0 a 1 γ κ − 3 α a 1 C 2 η 2 κ ℋ − a a 1 C 2 η 2 ℋ − 9 a a 0 κ 2 − 3 a 0 κ ν = 0 ,

a a − 1 3 β 3 α + 2 a a − 1 3 γ 3 α + 6 α a − 1 A 2 η 2 κ + 2 a a − 1 A 2 η 2 + 12 α a − 1 A B η 2 κ ℋ 2 + 4 a a − 1 A B η 2 ℋ 2 − 12 α a − 1 A C η 2 κ ℋ − 4 a a − 1 A C η 2 ℋ + a − 1 3 β κ + 6 α a − 1 B 2 η 2 κ ℋ 4 + 2 a a − 1 B 2 η 2 ℋ 4 − 12 α a − 1 B C η 2 κ ℋ 3 − 4 a a − 1 B C η 2 ℋ 3 + 2 a − 1 3 γ κ + 6 α a − 1 C 2 η 2 κ ℋ 2 + 2 a a − 1 C 2 η 2 ℋ 2 = 0 ,

a a − 1 2 a 0 β α + 2 a a − 1 2 a 0 γ α − 18 α a − 1 A B η 2 κ ℋ − 6 a a − 1 A B η 2 ℋ + 9 α a − 1 A C η 2 κ + 3 a a − 1 A C η 2 + 3 a − 1 2 a 0 β κ − 18 α a − 1 B 2 η 2 κ ℋ 3 − 6 a a − 1 B 2 η 2 ℋ 3 + 27 α a − 1 B C η 2 κ ℋ 2 + 9 a a − 1 B C η 2 ℋ 2 + 6 a − 1 2 a 0 γ κ − 9 α a − 1 C 2 η 2 κ ℋ − 3 a a − 1 C 2 η 2 ℋ = = 0 ,

− 2 a 2 a − 1 κ α + a a − 1 a 0 2 β α + a a − 1 2 a 1 β α + 2 a a − 1 a 0 2 γ α + 2 a a − 1 2 a 1 γ α − 9 α a − 1 κ 3 − a a − 1 ν α + 6 α a − 1 A B η 2 κ + 2 a a − 1 A B η 2 + 3 a − 1 a 0 2 β κ + 3 a − 1 2 a 1 β κ + 18 α a − 1 B 2 η 2 κ ℋ 2 + 6 a a − 1 B 2 η 2 ℋ 2 − 18 α a − 1 B C η 2 κ ℋ − 6 a a − 1 B C η 2 ℋ + 6 a − 1 a 0 2 γ κ + 6 a − 1 2 a 1 γ κ + 3 α a − 1 C 2 η 2 κ + a a − 1 C 2 η 2 − 9 a a − 1 κ 2 − 3 a − 1 κ ν = 0 ,

− 2 a 2 a 1 κ α + a a − 1 a 1 2 β α + a a 0 2 a 1 β α + 2 a a − 1 a 1 2 γ α + 2 a a 0 2 a 1 γ α − 9 α a 1 κ 3 − a a 1 ν α + 6 α a 1 A B η 2 κ + 2 a a 1 A B η 2 + 3 a − 1 a 1 2 β κ + 3 a 0 2 a 1 β κ + 18 α a 1 B 2 η 2 κ ℋ 2 + 6 a a 1 B 2 η 2 ℋ 2 − 18 α a 1 B C η 2 κ ℋ − 6 a a 1 B C η 2 ℋ + 6 a − 1 a 1 2 γ κ + 6 a 0 2 a 1 γ κ + 3 α a 1 C 2 η 2 κ + a a 1 C 2 η 2 − 9 a a 1 κ 2 − 3 a 1 κ ν = 0 ,

a a 0 a 1 2 β α + 2 a a 0 a 1 2 γ α + 3 a 0 a 1 2 β κ − 18 α a 1 B 2 η 2 κ ℋ − 6 a a 1 B 2 η 2 ℋ + 9 α a 1 B C η 2 κ + 3 a a 1 B C η 2 + 6 a 0 a 1 2 γ κ = 0 ,

a a 1 3 β 3 α + 2 a a 1 3 γ 3 α + a 1 3 β κ + 6 α a 1 B 2 η 2 κ + 2 a a 1 B 2 η 2 + 2 a 1 3 γ κ = 0 .

Solving the above system of equations, we obtain the following sets of solutions: set 1.

(16) a 0 = a 1 C − 2 a 1 B ℋ 2 B , a − 1 = 0 , β = − 2 ( a 1 2 γ + 3 α B 2 η 2 ) a 1 2 , a = − 6 α κ 2 + 4 α A B η 2 + α ( − C 2 ) η 2 − 2 ν 4 κ .

set 2.

(17) a 1 = 0 , a − 1 = − 2 a 0 ( − A − B ℋ 2 + C ℋ ) C − 2 B ℋ , a = − 6 α κ 2 + 4 α A B η 2 + α ( − C 2 ) η 2 − 2 ν 4 κ , β = − 4 a 0 2 γ − 12 α B 2 η 2 ℋ 2 + 12 α B C η 2 ℋ − 3 α C 2 η 2 2 a 0 2 .

By substituting the values from Eqs (16) and (17) in Eq. (5) and utilizing Eqs (7)–(11), we can determine U ( ξ ) . Subsequently, by substituting U ( ξ ) in Eq. (3), we can obtain the solutions of Eq. (1) as follows:

Class 1: When A B > 0 , C = 0 :

(18) ψ ( x , t ) = a 0 + a 1 ℋ + A B P cos ( A B ξ ) + Q sin ( A B ξ ) Q cos ( A B ξ ) − P sin ( A B ξ ) + a − 1 ℋ + A B P cos ( A B ξ ) + Q sin ( A B ξ ) Q cos ( A B ξ ) − P sin ( A B ξ ) − 1 exp ( i ( ζ + t ω + ( − κ ) x ) ) .

Class 2: When A B < 0 , C = 0 :

(19) ψ ( x , t ) = a 0 + a 1 ℋ − ∣ A B ∣ B P sinh ( 2 ∣ A B ∣ ξ ) + P cosh ( 2 ∣ A B ∣ ξ ) + Q P sinh ( 2 ∣ A B ∣ ξ ) + P cosh ( 2 ∣ A B ∣ ξ ) − Q + a − 1 ℋ − ∣ A B ∣ B P sinh ( 2 ∣ A B ∣ ξ ) + P cosh ( 2 ∣ A B ∣ ξ ) + Q P sinh ( 2 ∣ A B ∣ ξ ) + P cosh ( 2 ∣ A B ∣ ξ ) − Q − 1 exp ( i ( ζ + t ω + ( − κ ) x ) ) .

Class 3: When A = 0 , B ≠ 0 , C = 0 :

(20) ψ ( x , t ) = a 0 + a 1 ℋ − P B ( P ξ + Q ) + a − 1 ℋ − P B ( P ξ + Q ) − 1 exp ( i ( ζ + t ω + ( − κ ) x ) ) .

Class 4: When C 2 − 4 A B ≥ 0 , C ≠ 0 :

(21) ψ ( x , t ) = a 0 + a 1 ℋ − C 2 − 4 A B P cosh 1 2 ξ C 2 − 4 A B + Q sinh 1 2 ξ C 2 − 4 A B ( 2 B ) P sinh 1 2 ξ C 2 − 4 A B + Q cosh 1 2 ξ C 2 − 4 A B − C 2 B + a − 1 ℋ − C 2 − 4 A B P cosh 1 2 ξ C 2 − 4 A B + Q sinh 1 2 ξ C 2 − 4 A B ( 2 B ) P sinh 1 2 ξ C 2 − 4 A B + Q cosh 1 2 ξ C 2 − 4 A B − C 2 B − 1 exp ( i ( ζ + t ω + ( − κ ) x ) ) .

Class 5: When C 2 − 4 A B ≤ 0 , C ≠ 0 :

(22) ψ ( x , t ) = a 0 + a 1 ℋ − − C 2 + 4 A B P cos 1 2 ξ − C 2 + 4 A B − Q sin 1 2 ξ − C 2 + 4 A B ( 2 B ) P sin 1 2 ξ − C 2 + 4 A B + Q cos 1 2 ξ − C 2 + 4 A B − C 2 B + a − 1 ℋ − − C 2 + 4 A B P cos 1 2 ξ − C 2 + 4 A B − Q sin 1 2 ξ − C 2 + 4 A B ( 2 B ) P sin 1 2 ξ − C 2 + 4 A B + Q cos 1 2 ξ − C 2 + 4 A B − C 2 B − 1 exp ( i ( ζ + t ω + ( − κ ) x ) ) .

5 Graphical illustrations

In this section, we introduce some figures in the two-dimension (2D) and three-dimension (3D) about some solutions of NPDE by the ( ℋ + G ′ G 2 ) - expansion method. In Figure 1, we introduce the graph of (18) with (16) at α = 0.4 , a 1 = 0.2 , A = 1.5 , B = 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 0.01 , Q = 7 , and ℋ = 0.5 . The graph of (19) with (16) at α = 0.4 , a 1 = 0.2 , A = − 1.5 , B = 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 0.01 , Q = 7 , and ℋ = 0.5 is presented in Figure 2. In Figure 3, we introduce the graph of (21) with (16) at α = 0.4 , a 1 = 0.2 , A = − 1.5 , B = 0.5 , γ = 0.1 , C = 0.1 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 0.01 , Q = 7 , and ℋ = 0.5 . The graph of (21) with (16) at a 1 = 0.001 , α = 0.4 , A = 0.5 , B = 0.8 , γ = 0.6 , C = 0.6 , ζ = 0 , η = 0.001 , κ = 0.1 , ν = 0.3 , P = 0.001 , Q = 7 , and ℋ = 0.01 is presented in Figure 4. In Figure 5, we introduce the graph of (18) with (17) at α = 0.01 , a 0 = 0.0001 , A = 1.4 , B = 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 4 , ν = 0.8 , P = 4 , Q = 11 , and ℋ = 0.1 . The graph of (19) with (17) at α = 0.01 , a 0 = 0.01 , A = 1.1 , B = − 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 4 , ν = 0.5 , P = 4 , Q = 7 , ℋ = 0.1 is presented in Figure 6. In Figure 7, we introduce the graph of (21) with (17) at α = 0.4 , a 0 = 0.001 , A = − 0.5 , B = 0.7 , γ = 0.1 , C = 0.1 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 4 , Q = 2.5 , and ℋ = 0.5 . The graph of (21) with (17) at α = 0.4 , a 0 = 0.001 , A = 0.3 , B = 0.4 , γ = 0.1 , C = 0.5 , ζ = 0 , η = 0.3 , κ = 0.4 , ν = 0.3 , P = 9 , Q = 8 , and ℋ = 0.1 is presented in Figure 8. We give 2D, 3D, and contour plots of the derived solutions that illustrate several soliton types, including dark, bright, and singular traveling wave solutions in optical fibers.

$Figure 1 The graph of (18) with (16) at α = 0.4 \alpha =0.4 , a 1 = 0.2 {a}_{1}=0.2 , A = 1.5 A=1.5 , B = 0.5 B=0.5 , γ = 0.1 \gamma =0.1 , C = 0 C=0 , ζ = 0 \zeta =0 , η = 0.1 \eta =0.1 , κ = 0.3 \kappa =0.3 , ν = 0.3 \nu =0.3 , P = 0.01 P=0.01 , Q = 7 Q=7 , and ℋ = 0.5 {\mathcal{ {\mathcal H} }}=0.5 .$

Figure 1

The graph of (18) with (16) at α = 0.4 , a 1 = 0.2 , A = 1.5 , B = 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 0.01 , Q = 7 , and ℋ = 0.5 .

$Figure 2 The graph of (19) with (16) at α = 0.4 \alpha =0.4 , a 1 = 0.2 {a}_{1}=0.2 , A = − 1.5 A=-1.5 , B = 0.5 B=0.5 , γ = 0.1 \gamma =0.1 , C = 0 C=0 , ζ = 0 \zeta =0 , η = 0.1 \eta =0.1 , κ = 0.3 \kappa =0.3 , ν = 0.3 \nu =0.3 , P = 0.01 P=0.01 , Q = 7 Q=7 , and ℋ = 0.5 {\mathcal{ {\mathcal H} }}=0.5 .$

Figure 2

The graph of (19) with (16) at α = 0.4 , a 1 = 0.2 , A = − 1.5 , B = 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 0.01 , Q = 7 , and ℋ = 0.5 .

$Figure 3 The graph of (21) with (16) at α = 0.4 \alpha =0.4 , a 1 = 0.2 {a}_{1}=0.2 , A = − 1.5 A=-1.5 , B = 0.5 B=0.5 , γ = 0.1 \gamma =0.1 , C = 0.1 C=0.1 , ζ = 0 \zeta =0 , η = 0.1 \eta =0.1 , κ = 0.3 \kappa =0.3 , ν = 0.3 \nu =0.3 , P = 0.01 P=0.01 , Q = 7 Q=7 , and ℋ = 0.5 {\mathcal{ {\mathcal H} }}=0.5 .$

Figure 3

The graph of (21) with (16) at α = 0.4 , a 1 = 0.2 , A = − 1.5 , B = 0.5 , γ = 0.1 , C = 0.1 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 0.01 , Q = 7 , and ℋ = 0.5 .

$Figure 4 The graph of (21) with (16) at a 1 = 0.001 {a}_{1}=0.001 , α = 0.4 \alpha =0.4 , A = 0.5 A=0.5 , B = 0.8 B=0.8 , γ = 0.6 \gamma =0.6 , C = 0.6 C=0.6 , ζ = 0 \zeta =0 , η = 0.001 \eta =0.001 , κ = 0.1 \kappa =0.1 , ν = 0.3 \nu =0.3 , P = 0.001 P=0.001 , Q = 7 Q=7 , and ℋ = 0.01 {\mathcal{ {\mathcal H} }}=0.01 .$

Figure 4

The graph of (21) with (16) at a 1 = 0.001 , α = 0.4 , A = 0.5 , B = 0.8 , γ = 0.6 , C = 0.6 , ζ = 0 , η = 0.001 , κ = 0.1 , ν = 0.3 , P = 0.001 , Q = 7 , and ℋ = 0.01 .

$Figure 5 The graph of (18) with (17) at α = 0.01 \alpha =0.01 , a 0 = 0.0001 {a}_{0}=0.0001 , A = 1.4 A=1.4 , B = 0.5 B=0.5 , γ = 0.1 \gamma =0.1 , C = 0 C=0 , ζ = 0 \zeta =0 , η = 0.1 \eta =0.1 , κ = 4 \kappa =4 , ν = 0.8 \nu =0.8 , P = 4 P=4 , Q = 11 Q=11 , and ℋ = 0.1 {\mathcal{ {\mathcal H} }}=0.1 .$

Figure 5

The graph of (18) with (17) at α = 0.01 , a 0 = 0.0001 , A = 1.4 , B = 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 4 , ν = 0.8 , P = 4 , Q = 11 , and ℋ = 0.1 .

$Figure 6 The graph of (19) with (17) at α = 0.01 \alpha =0.01 , a 0 = 0.01 {a}_{0}=0.01 , A = 1.1 A=1.1 , B = − 0.5 B=-0.5 , γ = 0.1 \gamma =0.1 , C = 0 C=0 , ζ = 0 \zeta =0 , η = 0.1 \eta =0.1 , κ = 4 \kappa =4 , ν = 0.5 \nu =0.5 , P = 4 P=4 , Q = 7 Q=7 , and ℋ = 0.1 {\mathcal{ {\mathcal H} }}=0.1 .$

Figure 6

The graph of (19) with (17) at α = 0.01 , a 0 = 0.01 , A = 1.1 , B = − 0.5 , γ = 0.1 , C = 0 , ζ = 0 , η = 0.1 , κ = 4 , ν = 0.5 , P = 4 , Q = 7 , and ℋ = 0.1 .

$Figure 7 The graph of (21) with (17) at α = 0.4 \alpha =0.4 , a 0 = 0.001 {a}_{0}=0.001 , A = − 0.5 A=-0.5 , B = 0.7 B=0.7 , γ = 0.1 \gamma =0.1 , C = 0.1 C=0.1 , ζ = 0 \zeta =0 , η = 0.1 \eta =0.1 , κ = 0.3 \kappa =0.3 , ν = 0.3 \nu =0.3 , P = 4 P=4 , Q = 2.5 Q=2.5 , and ℋ = 0.5 {\mathcal{ {\mathcal H} }}=0.5 .$

Figure 7

The graph of (21) with (17) at α = 0.4 , a 0 = 0.001 , A = − 0.5 , B = 0.7 , γ = 0.1 , C = 0.1 , ζ = 0 , η = 0.1 , κ = 0.3 , ν = 0.3 , P = 4 , Q = 2.5 , and ℋ = 0.5 .

$Figure 8 The graph of (21) with (17) at α = 0.4 \alpha =0.4 , a 0 = 0.001 {a}_{0}=0.001 , A = 0.3 A=0.3 , B = 0.4 B=0.4 , γ = 0.1 \gamma =0.1 , C = 0.5 C=0.5 , ζ = 0 \zeta =0 , η = 0.3 \eta =0.3 , κ = 0.4 \kappa =0.4 , ν = 0.3 \nu =0.3 , P = 9 P=9 , Q = 8 Q=8 , and ℋ = 0.1 {\mathcal{ {\mathcal H} }}=0.1 .$

Figure 8

The graph of (21) with (17) at α = 0.4 , a 0 = 0.001 , A = 0.3 , B = 0.4 , γ = 0.1 , C = 0.5 , ζ = 0 , η = 0.3 , κ = 0.4 , ν = 0.3 , P = 9 , Q = 8 , and ℋ = 0.1 .

6 MIA

Modulation instability (MI) refers to a phenomenon that occurs in nonlinear optical systems, particularly in optical fibers. It is a process where small perturbations or fluctuations in an optical signal can grow rapidly, leading to the formation of multiple distinct pulses within the signal. This instability arises due to the interplay between the nonlinear effects in the medium and the dispersion of the signal. MI is used to display the behavior of the continuous waves over a long time of evolution.

We suppose the following plane-wave as solution of (1):

(23) ψ ( x , t ) = p 0 exp ( i ( δ x − t ω ) ) ,

where p 0 is the input signal, δ is the wave vector, and ω is the angular frequency. Substituting (23) in (1), we obtain the linear dispersion as follows:

(24) ω = δ 2 ( a − α δ ) + p 0 2 ( β δ − b ) .

To obtain the linearized expression, one introduces the tiny perturbations in the plane-wave

(25) ψ ( x , t ) = ( μ ( x , t ) + p 0 ) exp ( i ( δ x − t ω ) ) ,

where μ ( x , t ) is the perturbed amplitude. Substituting (24) and (25) in (1), we obtain the following equation:

(26) 2 a δ μ x ( x , t ) − i a μ x x ( x , t ) + p 0 2 ( − i ( b − β δ ) μ ( x , t ) − i ( b − β δ ) μ * ( x , t ) + ( β + γ ) μ x ( x , t ) + γ μ x * ( x , t ) ) − 3 α δ 2 μ x ( x , t ) + 3 i α δ μ x x ( x , t ) + α μ x x x ( x , t ) + μ t ( x , t ) = 0 ,

where μ * is the complex conjugate of μ ( x , t ) . While μ ( x , t ) is the small perturbation and expressed as follows:

(27) μ ( x , t ) = f 1 exp ( i ( K x − Ω t ) ) + f 2 exp ( − i ( K x − Ω t ) ) ,

where K is the real disturbance wave-number, Ω is the complex frequency, and f 1 , f 2 are the coefficients of linear combination. Substituting (27) in (26), we may have the following homogeneous equations:

(28) f 1 ( − a K ( 2 δ + K ) + α K 3 + 3 α δ K 2 + 3 α δ 2 K + Ω ) + p 0 2 ( b ( f 1 + f 2 ) − f 1 ( β δ + K ( β + γ ) ) − f 2 ( β δ + γ K ) ) = 0 , f 2 ( a K ( K − 2 δ ) + α K 3 − 3 α δ K 2 + 3 α δ 2 K + Ω ) − p 0 2 ( b ( f 1 + f 2 ) − β δ f 1 + γ f 1 K − β δ f 2 + β f 2 K + γ f 2 K ) = 0 .

We evaluate the determinate of the above system and setting to zero, we obtain two-order polynomial equation

(29) Ω 2 + Λ 1 Ω + Λ 0 = 0 ,

where

(30) Λ 0 = − a 2 K 4 + 4 a 2 δ 2 K 2 + 2 a b K 2 p 0 2 + 2 a α δ K 4 − 12 a α δ 3 K 2 + 2 a β δ K 2 p 0 2 + 4 a γ δ K 2 p 0 2 − 6 α b δ K 2 p 0 2 + α 2 K 6 − 3 α 2 δ 2 K 4 − 2 α β K 4 p 0 2 − 2 α γ K 4 p 0 2 + 9 α 2 δ 4 K 2 − 6 α γ δ 2 K 2 p 0 2 + β 2 K 2 p 0 4 + 2 β γ K 2 p 0 4 , Λ 1 = − 4 a δ K + 2 α K 3 + 6 α δ 2 K − 2 β K p 0 2 − 2 γ K p 0 2 ,

and the MI growth rate is as follows:

(31) σ = ∣ Im [ Ω max ] ∣ .

Considering (29), the following cases of the modulation instability of (1) can be discussed as follows:

(32) Ω = ∓ a 2 K 4 − 2 a b K 2 p 0 2 − 6 a α δ K 4 + 2 a β δ K 2 p 0 2 + 6 α b δ K 2 p 0 2 + 9 α 2 δ 2 K 4 − 6 α β δ 2 K 2 p 0 2 + γ 2 K 2 p 0 4 + 2 a δ K − α K 3 − 3 α δ 2 K + β K p 0 2 + γ K p 0 2 .

The stability of the steady state is determined by (32). When Ω has an imaginary part, then steady-state solution is unstable because the perturbation rises exponentially. But if the wave number Ω is real, the steady state is stable against small perturbations. Thus, the modulation instability of (32) takes place when

(33) a 2 K 4 − 2 a b K 2 p 0 2 − 6 a α δ K 4 + 2 a β δ K 2 p 0 2 + 6 α b δ K 2 p 0 2 + 9 α 2 δ 2 K 4 − 6 α β δ 2 K 2 p 0 2 + γ 2 K 2 p 0 4 < 0 .

Eventually, we attain the MI gain spectrum as

(34) G ( K ) = 2 I m ( Ω ) = 2 Im a 2 K 4 − 2 a b K 2 p 0 2 − 6 a α δ K 4 + 2 a β δ K 2 p 0 2 + 6 α b δ K 2 p 0 2 + 9 α 2 δ 2 K 4 − 6 α β δ 2 K 2 p 0 2 + γ 2 K 2 p 0 4 .

In the following figures, we show that modulation instability is significantly affected by changing the parameters.

Modulation instability refers to the phenomenon where small perturbations in an optical signal can grow exponentially, leading to the formation of multiple distinct pulses. This is particularly significant in optical fibers, where nonlinear effects interact with dispersion. Figures 9, 10, 11, 12, 13 collectively illustrate how modulation instability manifests in optical systems. Varying parameters like nonlinearity and dispersion can create different growth patterns, reinforcing the importance of understanding these interactions for applications in nonlinear optics and fiber communications.

$Figure 9 The 2D graph of (29) for the values of α = 1 \alpha =1 , a = 1 a=1 , b = 2 b=2 , β = 2.5 \beta =2.5 , δ = 0.9 \delta =0.9 , and γ = 1.8 \gamma =1.8 .$

Figure 9

The 2D graph of (29) for the values of α = 1 , a = 1 , b = 2 , β = 2.5 , δ = 0.9 , and γ = 1.8 .

$Figure 10 The 3D graph of (29) for the values of α = 1 \alpha =1 , a = 1 a=1 , b = 2 b=2 , β = 2.5 \beta =2.5 , δ = 0.9 \delta =0.9 , γ = 1.8 \gamma =1.8 with (a) p 0 = 2 {p}_{0}=2 , (b) p 0 = 2.5 {p}_{0}=2.5 , and (c) p 0 = 3 {p}_{0}=3 .$

Figure 10

The 3D graph of (29) for the values of α = 1 , a = 1 , b = 2 , β = 2.5 , δ = 0.9 , γ = 1.8 with (a) p 0 = 2 , (b) p 0 = 2.5 , and (c) p 0 = 3 .

$Figure 11 The graph of (34) for the values of α = 0.3 \alpha =0.3 , β = 0.5 \beta =0.5 , b = 3 b=3 , γ = 5 \gamma =5 , δ = 5 \delta =5 , and p 0 = 0.1 {p}_{0}=0.1 .$

Figure 11

The graph of (34) for the values of α = 0.3 , β = 0.5 , b = 3 , γ = 5 , δ = 5 , and p 0 = 0.1 .

$Figure 12 The graph of (34) for the values of α = 2 \alpha =2 , a = 1 a=1 , β = 2 \beta =2 , b = 2 b=2 , γ = 4 \gamma =4 , and δ = 1 \delta =1 .$

Figure 12

The graph of (34) for the values of α = 2 , a = 1 , β = 2 , b = 2 , γ = 4 , and δ = 1 .

$Figure 13 The graph of (34) for the values of α = 2 \alpha =2 , a = 1.5 a=1.5 , β = 1.3 \beta =1.3 , γ = 0.8 \gamma =0.8 , δ = 1.1 \delta =1.1 , and p 0 = 5 {p}_{0}=5 .$

Figure 13

The graph of (34) for the values of α = 2 , a = 1.5 , β = 1.3 , γ = 0.8 , δ = 1.1 , and p 0 = 5 .

7 Bifurcation analysis

In this section, we transform Eq. (13) into a planar dynamical system using bifurcation theory [37], to obtain the following system:

(35) U ′ ( ξ ) = W ( ξ ) , W ′ ( ξ ) = 1 A 1 ( A 2 U ( ξ ) − A 3 ( U ( ξ ) ) 3 ) ,

where A 1 = η 2 ( 3 α κ + a ) , A 2 = α κ 3 + a κ 2 + ω , and A 3 = β κ + b .

It is important to show that the set of differential Eq. (35) is a conservative system if ∇ → . ℱ → = 0 , such that ℱ → = W i → + 1 A 1 ( A 2 U − A 3 U 3 ) j → is a vector field and constitutes the Hamiltonian function. The Hamiltonian function represents the total energy of system (35) (kinetic energy and potential energy), can be defined as follows:

(36) H ( U , W ) = W 2 2 + 1 4 A 1 ( A 3 U 4 − 2 A 2 U 2 ) = h .

The dynamical system (35) with parameters A 1 , A 2 , A 3 in the U W − phase plane has three equilibrium points e i = ( U i , W ) , i = 1 , 2 , 3 , where U 1 = 0 , U 2 = A 2 A 3 , U 3 = − A 2 A 3 and W = 0 . We determine the Jacobian matrix determinant of system (35)

(37) D i = ∣ J ( U i , W ) ∣ = 1 A 1 ( 3 U i 2 A 3 − A 2 ) , i = 1 , 2 , 3 .

Now, we display four cases for the bifurcation of phase portraits as follows:

Case 1: If A 2 A 3 < 0 and A 1 A 2 > 0 , there exist one equilibrium point e 1 ( 0,0 ) of system (35) and D e 1 ( 0,0 ) < 0 , that is a saddle point that has homoclinic orbits as shown in Figure 14.

$Figure 14 (a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , and A 3 = − 2 {{\mathcal{A}}}_{3}=-2 .$

Figure 14

(a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = 3 , A 2 = 1 , and A 3 = − 2 .

Case 2: If A 2 A 3 < 0 and A 1 A 2 < 0 , obtain one equilibrium point e 1 ( 0,0 ) of system (35), and D e 1 ( 0,0 ) > 0 that is a center point that has periodic orbits (Figure 15).

$Figure 15 (a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = − 3 {{\mathcal{A}}}_{1}=-3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , and A 3 = − 2 {{\mathcal{A}}}_{3}=-2 .$

Figure 15

(a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = − 3 , A 2 = 1 , and A 3 = − 2 .

Case 3: If A 2 A 3 > 0 and A 1 A 2 > 0 , obtain three equilibrium points of system (35) as follows: (i) The saddle point e 1 ( 0,0 ) , has D e 1 ( 0,0 ) < 0 , (ii) the center points e 2 ( A 2 A 3 , 0 ) and e 3 ( − A 2 A 3 , 0 ) have positive signs of the Jacobian D 2,3 corresponding to e 2 ( A 2 A 3 , 0 ) and e 3 ( − A 2 A 3 , 0 ) , respectively. Consequently, Figure 17 shows that the trajectories of the equilibrium points that are superperiodic orbits.

$Figure 16 3D plots of (36): (a) A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = − 2 {{\mathcal{A}}}_{3}=-2 and (b) A 1 = − 3 {{\mathcal{A}}}_{1}=-3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = − 2 {{\mathcal{A}}}_{3}=-2 .$

Figure 16

3D plots of (36): (a) A 1 = 3 , A 2 = 1 , A 3 = − 2 and (b) A 1 = − 3 , A 2 = 1 , A 3 = − 2 .

Case 4: If A 2 A 3 > 0 and A 1 A 2 < 0 , acquire three equilibrium points of system (35) as follows: (i) the center point e 1 ( 0,0 ) , has D e 1 ( 0,0 ) > 0 , (ii) the saddle points e 2 ( A 2 A 3 , 0 ) and e 3 ( − A 2 A 3 , 0 ) , have negative signs of the Jacobian D 2,3 matching e 2 ( A 2 A 3 , 0 ) and e 3 ( − A 2 A 3 , 0 ) , respectively. So, Figure 18 exhibits the trajectories of the equilibrium points that are hetroclinic orbits. Then, all phase portraits and contour plots of system (35) are shown in 2D plots (Figures 14, 15, 17, and 18). While Figures 16 and 19 exhibit 3D plots of the total energy defined by the Hamiltonian function (36).

$Figure 17 (a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 .$

Figure 17

(a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = 3 , A 2 = 1 , A 3 = 2 .

$Figure 18 (a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = − 3 {{\mathcal{A}}}_{1}=-3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 .$

Figure 18

(a) Phase plot and (b) Contour plot of system (35) with parameters: A 1 = − 3 , A 2 = 1 , A 3 = 2 .

$Figure 19 3D plots of (36): (a) A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 and (b) A 1 = − 3 {{\mathcal{A}}}_{1}=-3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 .$

Figure 19

3D plots of (36): (a) A 1 = 3 , A 2 = 1 , A 3 = 2 and (b) A 1 = − 3 , A 2 = 1 , A 3 = 2 .

8 Chaotic analysis

In this section, we illustrate the chaotic behaviors of the dynamical system presented in (35). This study is offered for three cases of this system, the first for unexcited system, the second for excited system by a parametric excitation force, and the third for control excited system using proportional feedback control (P-Controller).

8.1 Chaotic behaviors for unexcited system

We use time histories and Poincare mapping to investigate the chaotic behaviors for unexcited system (35). Figures 20 and 21 present time histories and Poincare mapping of unexcited system for the cases, A 1 A 2 > 0 , A 3 A 2 > 0 , and A 1 A 2 < 0 , A 3 A 2 < 0 . From these figures, we can conclude that system (35) is stable as shown in time histories in Figures 20(a) and 21(a). Also, system (35) is quasi-periodic as shown by Poincare mapping in Figures 20(b) and 21(b).

$Figure 20 (a) Time history and (b) Poincare mapping of unexcited system (35) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , and A 3 = 2 {{\mathcal{A}}}_{3}=2 .$

Figure 20

(a) Time history and (b) Poincare mapping of unexcited system (35) with parameters: A 1 = 3 , A 2 = 1 , and A 3 = 2 .

$Figure 21 (a) Time history and (b) Poincare mapping of unexcited system (35) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = − 1 {{\mathcal{A}}}_{2}=-1 , and A 3 = 2 {{\mathcal{A}}}_{3}=2 .$

Figure 21

(a) Time history and (b) Poincare mapping of unexcited system (35) with parameters: A 1 = 3 , A 2 = − 1 , and A 3 = 2 .

8.2 Chaotic behaviors for excited system

We modulated the dynamical system presented in (35) by adding a parametric excitation force as follows:

(38) U ′ ( ξ ) = W ( ξ ) , W ′ ( ξ ) = A 2 A 1 ( U ( ξ ) ) − A 3 A 1 ( U ( ξ ) ) 3 + f U ( ξ ) cos ( χ ξ ) ,

where f and χ are the amplitude and the frequency of the parametric force, respectively. We investigated the periodic behavior of the excited dynamical system presented in (38) using time histories and Poincare mapping, as shown in Figures 22, 23, and 24. These figures show the influence of the parametric excitation force on the system which made the system unstable and chaotic. Moreover, the parametric force effects on sensitivity to the initial conditions of the dynamical system. Therefore, this system must be controlled using a different type of control.

$Figure 22 (a) Time history and (b) Poincare mapping of excited system (38) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = − 1 {{\mathcal{A}}}_{2}=-1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 , f = 0.1 f=0.1 , and χ = 0.5 \chi =0.5 .$

Figure 22

(a) Time history and (b) Poincare mapping of excited system (38) with parameters: A 1 = 3 , A 2 = − 1 , A 3 = 2 , f = 0.1 , and χ = 0.5 .

$Figure 23 (a) Time history and (b) Poincare mapping of excited system (38) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = − 1 {{\mathcal{A}}}_{2}=-1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 , f = 0.3 f=0.3 , and χ = 0.5 \chi =0.5 .$

Figure 23

(a) Time history and (b) Poincare mapping of excited system (38) with parameters: A 1 = 3 , A 2 = − 1 , A 3 = 2 , f = 0.3 , and χ = 0.5 .

$Figure 24 (a) Time history and (b) Poincare mapping of excited system (38) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = − 1 {{\mathcal{A}}}_{2}=-1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 , f = 0.3 f=0.3 , and χ = 0.2 \chi =0.2 .$

Figure 24

(a) Time history and (b) Poincare mapping of excited system (38) with parameters: A 1 = 3 , A 2 = − 1 , A 3 = 2 , f = 0.3 , and χ = 0.2 .

8.3 Chaotic behaviors for control excited system

To suppress the vibrations of the dynamical system presented in (38), we will use the proportional feedback control (P-Controller) to improve system stability. P-Controller can be represented mathematically by F ( ξ ) = − k U ( ξ ) , where F ( ξ ) is the output of the proportional controller, U ( ξ ) is the error signal, and k is the proportional gain parameter as shown in the block diagram in Figure 25. So, the system presented in (38) after using P-Controller takes the form:

(39) U ′ ( ξ ) = W ( ξ ) , W ′ ( ξ ) = A 2 A 1 ( U ( ξ ) ) − A 3 A 1 ( U ( ξ ) ) 3 + f U ( ξ ) c o s ( χ ξ ) + F ( ξ ) .

Figure 25

Closed loop of controlled system (39).

We study the controlled system (39) with the following values of parameters: A 1 = 3 , A 2 = 1 , A 3 A 2 = 2 , f = 0.3 , χ = 1.2 , and different values of the proportional gain parameter. For k = 0 (P-Controller deactivated), the dynamical system (39) has high amplitude and still unstable with chaotic behavior as presented in Figure 26. When the P-Controller activated the amplitude of the dynamical system (39) decreased as shown in Figures 27 and 28. In Figure 27, we take k = 0.5 ; so, the amplitude decreases to 0.02, which means that the effectiveness of P-Controller E a = amplitude without controller amplitude with controller equals 60 but the system will still be unstable and chaotic. In Figure 28, by increasing the value of k to 2.5, the amplitude decreases to 0.007, which means that E a = 171 and system (39) becomes stable with limit cycle phase.

$Figure 26 (a) Time history and (b) Poincare mapping of excited system (39) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 , f = 0.3 f=0.3 , χ = 1.2 \chi =1.2 , and k = 0 k=0 .$

Figure 26

(a) Time history and (b) Poincare mapping of excited system (39) with parameters: A 1 = 3 , A 2 = 1 , A 3 = 2 , f = 0.3 , χ = 1.2 , and k = 0 .

$Figure 27 (a) Time history and (b) Poincare mapping of excited system (39) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 , f = 0.3 f=0.3 , χ = 1.2 \chi =1.2 , and k = 0.5 k=0.5 .$

Figure 27

(a) Time history and (b) Poincare mapping of excited system (39) with parameters: A 1 = 3 , A 2 = 1 , A 3 = 2 , f = 0.3 , χ = 1.2 , and k = 0.5 .

$Figure 28 (a) Time history and (b) Poincare mapping of excited system (39) with parameters: A 1 = 3 {{\mathcal{A}}}_{1}=3 , A 2 = 1 {{\mathcal{A}}}_{2}=1 , A 3 = 2 {{\mathcal{A}}}_{3}=2 , f = 0.3 f=0.3 , χ = 1.2 \chi =1.2 , and k = 2.5 k=2.5 .$

Figure 28

(a) Time history and (b) Poincare mapping of excited system (39) with parameters: A 1 = 3 , A 2 = 1 , A 3 = 2 , f = 0.3 , χ = 1.2 , and k = 2.5 .

9 Conclusion

In this study, we have explored the optical soliton solutions and bifurcation analysis of the Sasa–Satsuma equation, emphasizing its chaotic behaviors and modulation instability within nonlinear optical systems. Utilizing the ( ℋ + G ′ G 2 ) -expansion method, we derived analytical solutions that illustrate the complex dynamics of solitons in optical fibers. Our analysis of modulation instability revealed the critical role of higher-order nonlinear effects, which can significantly influence the stability of optical signals. We conducted a thorough bifurcation analysis, employing time histories and Poincaré mapping to investigate the chaotic behaviors induced by parametric excitation. Additionally, we implemented a proportional feedback control (P-Controller) to manage the vibrations of the dynamical system effectively. The graphical representations provided in both 2D and 3D formats enhance our understanding of the soliton solutions and their behaviors under various conditions. Overall, our findings contribute valuable insights into the dynamics of NPDEs, with implications for advancements in nonlinear optics and related fields. Future work may focus on further exploring these phenomena under different parameters and conditions, as well as applying the methodologies to other nonlinear systems.

Acknowledgments

The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-73).

Funding information: The authors extend their appreciation to Taif University, Saudi Arabia, for supporting this work through project number (TU-DSPP-2024-73).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Received: 2024-11-29

Revised: 2025-03-18

Accepted: 2025-04-15

Published Online: 2025-06-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/phys-2025-0164

Schlagwörter für diesen Artikel

Sasa–Satsuma equation; (ℋ + G′/G2)-expansion method; nonlinear optical system; qualitative analysis; dynamical system; P-controller

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