Analytical exploration and parametric insights into optical solitons in magneto-optic waveguides: Advances in nonlinear dynamics for applied sciences

Dean Chou; Hamood Ur Rehman; Kiran Khushi; Ifrah Iqbal

doi:10.1515/nleng-2025-0166

Artikel Open Access

Analytical exploration and parametric insights into optical solitons in magneto-optic waveguides: Advances in nonlinear dynamics for applied sciences

Dean Chou , Hamood Ur Rehman , Kiran Khushi und Ifrah Iqbal

Veröffentlicht/Copyright: 3. November 2025

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Abstract

Within this study, we delve into novel optical solitons concerning the perturbed nonlinear Schrödinger equation, which governs the propagation of electromagnetic waves in magneto-optic waveguides. These components play vital roles in optical transmission lines and lasers, serving critical functions within integrated optical circuits as circulators, modulators, and isolators. Employing the simplest equation method, the 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method, and the generalized Riccati equation mapping method, we derive single and multi-optical soliton solutions. Our analysis yields a diverse array of solutions, including multi-soliton, singular, combo bright-dark, periodic singular, bright, and dark optical solitons. Additionally, we discuss the parametric conditions crucial for both the existence and shaping of these solitons. The visual representation of our findings through three-dimensional, two-dimensional, and contour plots elucidates the dynamic phenomena and interprets the physical implications of these solutions using various parameter values. This research demonstrates the efficacy of rational analytical methods in elucidating the intricate dynamics of soliton solutions in nonlinear optical systems, thereby offering valuable insights for further exploration in the fields of physics, mathematics, sciences, and engineering.

Keywords: optical solitons; nonlinear Schrödinger equation; analytical methods; parametric analysis; magneto-optic waveguides

1 Introduction

Nonlinear partial differential equations (NLPDEs) find applications across various scientific domains including oceanography, mechanics, aeronautics, nonlinear optics, plasma physics, and many others [1–5]. These equations emerge from diverse mathematical and physical models, each carrying significant implications in the real world. Among the myriad phenomena encountered in NLPDE solutions [6–14], solitons emerge as fundamental elements in the exploration of nonlinear processes. Their pervasive presence extends to fields such as quantum electronics, physics, engineering, and fluid dynamics, offering crucial insights into the behaviour of nonlinear media [15–20]. The quest for exact soliton solutions to NLPDEs is paramount for understanding their mathematical and practical applications, thus attracting recent attention in the applied sciences and engineering community. Over the past few decades, several efficient methods have been developed to obtain exact solutions in the form of solitary waves, including the extended tanh–coth method [21], the homogeneous balance method [22], the simplest equation (SE) method (SEM) [5,23–25], the generalized Riccati equation mapping method (GREMM) [26], and many others [27–33].

For more than 40 years, researchers in various branches of applied mathematics have been studying the nonlinear Schrödinger Equation (NLSE) [34]. This equation is expressed as

(1) i u t + 1 2 u x x + ∣ u ∣ 2 u = 0 .

It has been proven to accurately capture the transmission of picosecond pulses in optical fibres. However, recent experimental advancements in the production of ultra-short femtosecond optical pulses have revealed limitations in the NLSE’s result as pulse duration decreases [35]. Consequently, a higher-order NLSE becomes necessary. Kodama [36] proposed a perturbed nonlinear Schrödinger equation utilizing a multiple-scale perturbation by applying Maxwell’s equation for an electric field in a fibre with an inhomogeneous dielectric constant:

(2) i u t + 1 2 u x x + ∣ u ∣ 2 u = i σ ( β 1 u x x x + β 2 ∣ u ∣ 2 u x + β 3 u 2 u x ∗ ) − i Δ u ,

where β i , i = 1 , 2 , 3 , and Δ , are constants, u = u ( x , t ) is a complex function with the complex conjugate denoted by u ∗ , u x x represents the group velocity dispersion, u x x x denotes the third-order dispersion, ∣ u ∣ 2 u , ∣ u ∣ 2 u x , and u 2 u x ∗ are nonlinear terms with ∣ u ∣ 2 u representing the Kerr law, and Δ is a small parameter ( Δ ≪ 1 ) . Additionally, the terms proportional to β i in Eq. (2) show higher-order dispersion effects, along β 1 represents third-order dispersion while β 2 and β 3 indicate higher dispersion effects. A more general form of Eq. (2) with Δ = 0 is expressed as

(3) i u t + α 1 u x x + α 2 ∣ u ∣ 2 u + i α 3 ( β 1 u x x x + β 2 ∣ u ∣ 2 u x + β 3 u 2 u x ∗ ) = 0 ,

where α i , i = 1 , 2 , 3 , are constants that control the strength of nonlinear effects, linear dispersion, and higher-order dispersion. Setting α 1 = 1 2 , α 2 = 1 , and α 3 = β 1 = β 2 = β 3 = 0 in Eq. (3) yields the NLSE as given in Eq. (1). Similarly, setting α 1 = 1 2 , α 2 = β 1 = 1 , α 3 = τ , and β 2 = β 3 = 0 in Eq. (3) yields the nonlinear Schrödinger-Hirota equations as previously shown [35–37]:

(4) i u t + 1 2 u x x + ∣ u ∣ 2 u + i τ u x x x = 0 .

The propagation of optical solitons in dispersive optical fibres is governed by this equation. Eq. (3) finds application in nonlinear optics and biological phenomena. It is also referred to as the generalized Davydov equation, derived from the study of the nonlinear dynamics of α -helical proteins in DNA molecules in the field of biology when α 1 = α 2 = β 1 = 1 [38]. Moreover, Eq. (3) describes the propagation of femtosecond pulses in a monomode optical fibre as a model of a high-bit-rate, long-distance transmission system in the context of nonlinear optics. Hosseini et al. [39] employed the modified Jacobi elliptic expansion method to generate solitons and Jacobi elliptic solutions, while Kumari et al. [40] applied the generalized Jacobi elliptic method to obtain Jacobi elliptic-type solutions [41].

In this article, researchers have shown a keen interest in the study of multi-soliton solutions of fully integrable equations in solitary wave theory. First, we elaborate on the use of SEM and investigate an intriguing relationship between Burger’s equation, multi-soliton solutions, and the SEs [42]. The 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method is predominantly applicable to studying embedded solitons, and the current approach is not necessarily robust [43]. GREMM is employed in this study to obtain solitary waves and other types of solutions. Many solutions are obtained as a result, including periodic function, dark-soliton, singular soliton, bright soliton, and combined dark-bright soliton. We use the GREMM to acquire solitary waves and other forms of solutions [44]. The GREMM is a powerful analytical technique for solving a wide range of differential equations, particularly nonlinear ones. It has several advantages, but it also comes with some limitations and potential disadvantages. The GREMM is a valuable analytical tool for solving a broad class of nonlinear differential equations. Its advantages include generality, nonlinearity handling, and the potential to reduce problems to simpler forms. Our findings could be crucial for further understanding wave processes because the generalized unstable NLSE contains elements that could explain waves in the ocean and optical fibres more precisely than the NLSE [45].

We compared our results with previous studies, specifically the dark, periodic soliton solutions and dynamical phase portraits analysis of perturbed NLSE using the modified Sardar sub-equation approach and bifurcation analysis [46]. Various soliton solutions, such as breather, lump, singular, and periodic, were obtained by applying the tanh–coth method and the energy balance method [47]. The generalized Kudryashov method was used to extract soliton solutions comprising hyperbolic, exponential, trigonometric, and other functions [48]. We employed the SEM, 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method, and the GREMM on perturbed NLSE to obtain periodic function, dark-soliton, singular soliton, bright soliton, and combined dark-bright soliton solutions.

In recent years, the study of NLPDEs has garnered significant attention due to their applicability in various fields such as fluid dynamics, optical fibres, and quantum mechanics. However, certain equations within this domain remain underexplored, particularly in the context of soliton solutions. In this work, we introduce a novel approach by employing the SEM 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method and the GREMM to solve a specific nonlinear equation. These methods have not previously been applied to this equation, making our approach unique. Moreover, our study is the first to derive multi-soliton solutions for this equation, filling a critical gap in the existing literature and offering new insights into the underlying dynamics. This advancement not only demonstrates the versatility of these methods but also opens avenues for further exploration in related equations and physical models.

The remainder of the article is structured as follows: Section 2 provides a mathematical analysis of the equation. Section 3 delves into the SEM by outlining its methodology and showcasing the results through its application. In Section 4, the 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method is discussed and applied within the context of the equation. Section 5 discusses GREMM in detail, highlighting its specific application within the context of the equation. Section 6 explaines the results and discussions, and finally, in Section 7, conclusion is presented.

2 Mathematical analysis

Suppose, the following transformation:

(5) u ( x , t ) = P ( δ ) e i ( v x + c t ) , δ = k x + ω t ,

where k and ω are arbitrary constants. By using the above transformation, we obtain the real and imaginary parts of Eq. (3) as follows:

(6) ( α 1 k 2 − 3 α 3 β 1 k 2 v ) P ″ ( δ ) + ( α 3 β 1 v 3 − α 1 v 2 − c ) P ( δ ) + ( α 2 − α 3 β 2 v + α 3 β 3 v ) P 3 ( δ ) = 0

and

(7) α 3 β 1 k 3 P ‴ ( δ ) + ( ω − 3 α 3 β 1 k v 2 + 2 α 1 k v ) P ′ ( δ ) + ( α 3 β 2 k + α 3 β 3 k ) P 2 ( δ ) P ′ ( δ ) = 0 .

Integrating Eq. (7) once w.r.t δ , we obtain

(8) α 3 β 1 k 3 P ″ ( δ ) + ( ω − 3 α 3 β 1 k v 2 + 2 α 1 k v ) P ( δ ) + 1 3 ( α 3 β 2 k + α 3 β 3 k ) P 3 ( δ ) = 0 .

3 SE method

Let NLPDE, we can reduce this equation to an ordinary differential equation:

(9) G ( u , u t , u x , u x x , … ) ,

(10) H ( P ( δ ) , d P d δ , d 2 P d δ 2 , … ) ,

The exact solution of Eq. (10) can be written as

(11) P ( δ ) = ∑ j = 0 N a j ( Y ( δ ) ) j ,

where a j ( j = 0 , 1 , … , N ) are constants to be determined, Y is the exact solution to an ODE known as the SE. N is obtained by using the homogeneous balancing rule.

The application steps of the SEM as follows:

Using the traveling wave ansatz, the NLPDE is reduced to an ODE as given in Eq. (10).
Using the balance equation between the highest order nonlinear term and the highest-order derivative found in Eq. (10), the value of N can be determined.
Put the finite-series solution Eq. (11) in Eq. (10) and as a result a polynomial of Y is obtained.
Equating all the coefficients to zero yields a system that can be solved to find a j . The determination of the solution for Eq. (11) is completed by substituting the values of a j into Eq. (10).

Hence, the above-mentioned method is only used to obtain single solitary solutions and does not work for finding multi-soliton solutions. Burger’s equation is the SE, which builds multi-soliton solutions for the nonlinear equations.

Burger’s equation

In order to investigate multi-soliton solutions of the proposed equation, Burger’s equation is chosen as the SE as it is fully integrable in ( 1 + 1 ) -dimensional. Consider Eq. (12)

(12) u t − 2 u u x − u x x = 0 ,

where γ = − 2 is the arbitrary constant. The traveling wave ansatz is δ = k x + ω t , where k is the wave number and ω is the wave speed. Eq. (12) is transformed into

(13) ω Y δ − 2 k Y Y δ − k 2 Y δ δ = 0 .

Integrating Eq. (13) once with respect to δ , it becomes

(14) Y δ = ω k 2 Y − k k 2 Y 2 .

It is commonly known that the dispersion relation of Eq. (12) is

(15) ω = k 2 .

Hence, Eq. (14) is rewritten as

(16) Y δ = Y − 1 k Y 2 .

Thus, by using Hirota’s method, the general form of multi-soliton solution of Eq. (16) is derived as:

(17) Y = ∑ j = 1 N k j e k j x + ω j t 1 + ∑ j = 1 N e k j x + ω j t ,

where k j and ω j are the arbitrary constants.

3.1 Application of SEM

By balancing P ″ ( δ ) and P 3 ( δ ) in Eq. (8), we obtain N = 1 .

(18) P ( δ ) = a 0 + a 1 Y ( δ ) , a 1 ≠ 0 .

Put Eqs. (18) and (16) in Eq. (8). Then, by equating the coefficients of Y 0 ( δ ) , Y 1 ( δ ) , Y 2 ( δ ) , Y 3 ( δ ) and setting them to zero, we have following system of equations:

(19) 2 a 0 k v α 1 − 3 a 0 k v 2 α 3 β 1 + 1 3 a 0 3 k α 3 β 2 + 1 3 a 0 3 k α 3 β 3 + a 0 ω = 0 , 2 a 1 k v α 1 − 3 a 1 k v 2 α 3 β 1 + a 0 2 a 1 k α 3 β 2 + a 0 2 a 1 k α 3 β 3 + a 1 k 3 α 3 β 1 + a 1 ω = 0 , a 0 a 1 2 k α 3 β 2 + a 0 a 1 2 k α 3 β 3 − 3 a 1 k 2 α 3 β 1 = 0 , 2 a 1 k α 3 β 1 + 1 3 a 1 3 k α 3 β 2 + 1 3 a 1 3 k α 3 β 3 = 0 .

Solving Eq. (19), we obtain the following values of constants:

(20) a 0 = a 0 , a 1 = − 2 a 0 k , ω = − 6 v k 2 α 1 − 6 v 2 a 0 2 α 3 β 2 − a 0 2 k 2 α 3 β 2 − 6 v 2 a 0 2 α 3 β 3 − a 0 2 k 2 α 3 β 3 3 k , β 1 = − 2 a 0 2 ( β 1 + β 2 ) 3 k 2 .

Now, by substituting the above values in Eq. (18) then in Eq. (5), we have the following solutions:

(21) u 1 , * ( x , t ) = a 0 − 2 a 0 k j ∑ j = 1 N e ( k j x + ω j t ) k j 1 + ∑ j = 1 N e ( k j x + ω j t ) e i ( v x + c t ) .

For 1-soliton solution, we have

(22) u 1,1 ( x , t ) = a 0 − 2 a 0 e ( k 1 x + ω 1 t ) ( 1 + e ( k 1 x + ω 1 t ) ) e i ( v x + c t ) .

For 2-soliton solution, we have

(23) u 1,2 ( x , t ) = a 0 − 2 a 0 e ( k 1 x + ω 1 t ) + e ( k 2 x + ω 2 t ) ( 1 + e ( k 1 x + ω 1 t ) + e ( k 2 x + ω 2 t ) ) e i ( v x + c t ) .

For 3-soliton solution, we have

(24) u 1,3 ( x , t ) = a 0 − 2 a 0 e ( k 1 x + ω 1 t ) + e ( k 2 x + ω 2 t ) + e ( k 3 x + ω 3 t ) ( 1 + e ( k 1 x + ω 1 t ) + e ( k 2 x + ω 2 t ) + e ( k 3 x + ω 3 t ) ) × e i ( v x + c t ) .

4 Description of 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method

Let the solution of Eq. (8) be

(25) Y ( δ ) = q 0 + ∑ i = 1 N q i + s i φ ′ ( δ ) i φ ( δ ) i ,

where q 0 , q i , and s i ( i = 1 , 2 … N ) are constants. N can be obtained by using the homogeneous balancing rule and φ ( δ ) represents the following ODE:

(26) φ ′ ( δ ) 2 = φ ( δ ) 2 − d ,

where

(27) φ ( δ ) = b e δ + d 4 b e δ .

Now, by inserting Eq. (25) along Eq. (26) into Eq. (8), the system of equations is attained and by solving it, we obtain the values of constants.

4.1 Application of 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method

Now, by using the homogeneous balance rule, we obtain N = 1 .

(28) Y ( δ ) = q 0 + q 1 + s 1 φ ′ ( δ ) φ ( δ ) .

Now, by using Eq. (28) along Eqs. (26) and (27) into Eq. (8), the system of equations is attained

(29) α 3 β 1 k 3 q 0 + 1 3 α 3 β 2 k q 0 3 + 1 3 α 3 β 3 k q 0 3 + α 3 β 2 k q 0 s 1 2 + α 3 β 3 k q 0 s 1 2 − 3 α 3 β 1 k q 0 v 2 + 2 α 1 k q 0 v + q 0 ω = 0 , α 3 β 2 k q 1 3 + α 3 β 3 k q 1 3 − 3 α 3 β 2 k q 1 s 1 2 d − 3 α 3 β 3 k q 1 s 1 2 d = 0 , α 3 β 2 k q 0 q 1 2 + α 3 β 3 k q 0 q 1 2 − α 3 β 2 k q 0 s 1 2 d − α 3 β 3 k q 0 s 1 2 d = 0 , α 3 β 1 k 3 q 1 + α 3 β 2 k q 0 2 q 1 + α 3 β 3 k q 0 2 q 1 + α 3 β 2 k q 1 s 1 2 + α 3 β 3 k q 1 s 1 2 − 3 α 3 β 1 k q 1 v 2 + 2 α 1 k q 1 v + q 1 ω = 0 , 3 α 3 β 2 k q 1 2 s 1 + 3 α 3 β 3 k q 1 2 s 1 + α 3 β 2 ( − k ) d s 1 3 − α 3 β 3 k d s 1 3 = 0 , α 3 β 2 k q 0 q 1 s 1 + 2 α 3 β 3 k q 0 q 1 s 1 = 0 , 3 α 3 β 1 k 3 s 1 + 3 α 3 β 2 k q 0 2 s 1 + 3 α 3 β 3 k q 0 2 s 1 + α 3 β 2 k s 3 + α 3 β 3 k s 1 3 − 9 α 3 β 1 k s 1 v 2 + 6 α 1 k s 1 v + 3 s 1 ω = 0 .

and by solving it, we obtain the following values of constants.

Set 1:

q 0 = q 0 , q 1 = q 1 , β 2 = − β 3 , s 1 = 0 , α 3 = − 2 α 1 k v − ω β 1 k ( k 2 − 3 v 2 ) .

By putting these values in Eq. (28), we have

u 2 , * ( x , t ) = 4 b q 1 e δ 4 b 2 e 2 δ + d + q 0 e i ( v x + c t ) .

By taking d = ± 4 b 2 , we obtain

(30) u 2,1 ( x , t ) = q 0 + q 1 sech ( δ ) 2 b e i ( v x + c t ) .

(31) u 2,2 ( x , t ) = q 0 + q 1 csch ( δ ) 2 b e i ( v x + c t ) .

Set 2:

q 0 = 0 , q 1 = 0 , s 1 = s 1 β 2 = − β 3 , α 3 = − 2 α 1 k v − ω β 1 k ( k 2 − 3 v 2 ) .

By putting these values in Eq. (28), we have

u 2 , * * ( x , t ) = s 1 ( 4 b 2 e 2 δ − d ) 4 b 2 e 2 δ + d e i ( v x + c t ) .

By taking d = ± 4 b 2 , we obtain

(32) u 2,3 ( x , t ) = ( s 1 tanh ( δ ) ) e i ( v x + c t ) .

(33) u 2,4 ( x , t ) = ( s 1 coth ( δ ) ) e i ( v x + c t ) .

5 Generalized Riccati equation mapping method

Suppose, in Eq. (11), Y ( δ ) satisfies the following ODE:

(34) Y ′ ( δ ) = q + r ψ ( δ ) + s ψ 2 ( δ ) ,

where, q , r , and s are constants.

Type: 1 For r 2 − 4 q s > 0 and r s ≠ 0 or q r ≠ 0 , then

(35) Y 1 ( δ ) = − 1 2 s r + r 2 − 4 q s tanh r 2 − 4 q s 2 ( δ ) ,

(36) Y 2 ( δ ) = − 1 2 s r + r 2 − 4 q s coth r 2 − 4 q s 2 ( δ ) ,

(37) Y 3 ( δ ) = − 1 2 s ( r + r 2 − 4 q s ( tanh ( ( r 2 − 4 q s ) ( δ ) ) ± sech ( ( r 2 − 4 q s ) ( δ ) ) ) ) ,

(38) Y 4 ( δ ) = − 1 2 s ( r + r 2 − 4 q s ( coth ( ( r 2 − 4 q s ) ( δ ) ) ± csc h ( ( r 2 − 4 q s ) ( δ ) ) ) ) ,

(39) Y 5 ( δ ) = 1 4 s 2 r + r 2 − 4 q s tanh r 2 − 4 q s 4 ( δ ) ± coth r 2 − 4 q s 4 ( δ ) ,

(40) Y 6 ( δ ) = 1 2 s − r + ( C 2 + D 2 ) ( r 2 − 4 q s ) − C r 2 − 4 q s cosh ( r 2 − 4 q s ( δ ) ) C sinh ( r 2 − 4 q s ( δ ) ) + D ,

(41) Y 7 ( δ ) = 1 2 s − r − ( C 2 + D 2 ) ( r 2 − 4 q s ) + C r 2 − 4 q s cosh ( r 2 − 4 q s ( δ ) ) C sinh ( r 2 − 4 q s ( δ ) ) + D ,

where C and D are two non-zero real constants and satisfied D 2 − C 2 > 0 .

(42) Y 8 ( δ ) = 2 s cosh ( r 2 − 4 q s ( δ ) ) r 2 − 4 q s sinh r 2 − 4 q s 2 ( δ ) − r cosh r 2 − 4 q s 2 ( δ ) ,

(43) Y 9 ( δ ) = − 2 s sinh ( r 2 − 4 q s ( δ ) ) r sinh r 2 − 4 q s 2 ( δ ) − r 2 − 4 q s cosh r 2 − 4 q s 2 ( δ ) ,

(44) Y 10 ( δ ) = 2 s cosh r 2 − 4 q s 2 ( δ ) r 2 − 4 q s sinh ( r 2 − 4 q s ( δ ) ) − r cosh ( r 2 − 4 q s ( δ ) ) ± i r 2 − 4 q s ,

(45) Y 11 ( δ ) = 2 s cosh r 2 − 4 q s 2 ( δ ) − r sinh ( r 2 − 4 q s ( δ ) ) + r 2 − 4 q s cosh ( r 2 − 4 q s ) ± r 2 − 4 q s ,

(46) Y 12 ( δ ) = 4 s sinh 4 q s − r 2 4 ( δ ) cosh 4 q s − r 2 4 ( δ ) − 2 r sinh 4 q s − r 2 4 ( δ ) cosh 4 q s − r 2 4 ( δ ) + 2 4 q s − r 2 cosh 2 4 q s − r 2 4 ( δ ) − 4 q s − r 2 .

Type: 2 For r 2 − 4 q s < 0 and r s ≠ 0 or q r ≠ 0 , then

(47) Y 13 ( δ ) = 1 2 s − r + 4 q s − r 2 tan 4 q s − r 2 2 ( δ ) ,

(48) Y 14 ( δ ) = − 1 2 s r + 4 q s − r 2 cot 4 q s − r 2 2 ( δ ) ,

(49) Y 15 ( δ ) = 1 2 s ( − r + 4 q s − r 2 ( tan ( ( 4 q s − r 2 ) ( δ ) ) ± sec ( ( 4 q s − r 2 ) ( δ ) ) ) ) ,

(50) Y 16 ( δ ) = − 1 2 s ( r + 4 q s − r 2 ( cot ( ( 4 q s − r 2 ) ( δ ) ) ± csc ( ( 4 q s − r 2 ) ( δ ) ) ) ) ,

(51) Y 17 ( δ ) = 1 4 s − 2 r + 4 q s − r 2 tan 4 q s − r 2 4 ( δ ) − cot 4 q s − r 2 4 ( δ ) ,

(52) Y 18 ( δ ) = 1 2 s − r + ( C 2 + D 2 ) ( 4 q s − r 2 ) − C 4 q s − r 2 cos ( 4 q s − r 2 ( δ ) ) C sin ( 4 q s − r 2 ( δ ) ) + D ,

(53) Y 19 ( δ ) = 1 2 s − r − ( C 2 + D 2 ) ( 4 q s − r 2 ) + C r 2 − 4 q s cos ( 4 q s − r 2 ( δ ) ) C sin ( 4 q s − r 2 ( δ ) ) + D ,

where C and D are two non-zero real constants and satisfied C 2 − D 2 > 0 .

(54) Y 20 ( δ ) = − 2 s cos ( 4 q s − r 2 ( δ ) ) 4 q s − r 2 sin 4 q s − r 2 2 ( δ ) + q cos 4 q s − r 2 2 ( δ ) ,

(55) Y 21 ( δ ) = − 2 s sin ( 4 q s − r 2 ( δ ) ) − q sin 4 q s − r 2 2 ( δ ) + 4 q s − r 2 cos 4 q s − r 2 2 ( δ ) ,

(56) Y 22 ( δ ) = − 2 s cos 4 q s − r 2 2 ( δ ) 4 q s − r 2 sin ( 4 q s − r 2 ( δ ) ) + q cos ( 4 q s − r 2 ( δ ) ) ± i 4 q s − r 2 ,

(57) Y 23 ( δ ) = 2 s cos 4 q s − r 2 2 ( δ ) − q sin ( 4 q s − r 2 ( δ ) ) + 4 q s − r 2 cos ( 4 q s − r 2 ) ± 4 q s − r 2 ,

(58) Y 24 ( δ ) = 4 s sin 4 q s − r 2 4 ( δ ) cos 4 q s − r 2 4 ( δ ) − 2 r sin 4 q s − r 2 4 ( δ ) cos 4 q s − r 2 4 ( δ ) + 2 4 q s − r 2 cos 2 4 q s − r 2 4 ( δ ) − 4 q s − r 2 .

Type: 3 If s = 0 and q r ≠ 0 , then

(59) Y 25 ( δ ) = − r z q ( z + cosh ( q ( δ ) ) − sinh ( q ( δ ) ) ) ,

(60) Y 26 ( δ ) = − r ( cosh ( q ( δ ) ) + sinh ( q ( δ ) ) ) q ( z + cosh ( q ( δ ) ) + sinh ( q ( δ ) ) ) ,

where z is the arbitrary constant.

Type: 4 If q ≠ 0 and s = r = 0 , then

(61) Y 27 ( δ ) = − 1 q ( δ ) + l 1 ,

where l 1 is the arbitrary constant.

5.1 Application of GREMM

By balancing P ″ ( δ ) and P 3 ( δ ) in Eq. (8), we obtain N = 1 . We can express the solution of Eq. (11) as

(62) P ( δ ) = a 0 + a 1 Y ( δ ) , a 1 ≠ 0 .

Put Eq. (62) and Eq. (34) in Eq. (8). By equating the coefficients of Y 0 ( δ ) , Y 1 ( δ ) , Y 2 ( δ ) , Y 3 ( δ ) and setting them to zero, we obtain

(63) a 0 ω + 2 a 0 k v α 1 + q r a 1 k 3 α 3 β 1 − 3 a 0 k v 2 α 3 β 1 + 1 3 a 0 3 k α 3 β 2 + 1 3 a 0 3 k α 3 β 3 = 0 , a 1 ω + 2 a 1 k v α 1 + r 2 a 1 k 3 α 3 β 1 + 2 q s a 1 k 3 α 3 β 1 − 3 a 1 k v 2 α 3 β 1 + a 0 2 a 1 k α 3 β 2 + a 0 2 a 1 k α 3 β 3 = 0 , 3 r s a 1 k 3 α 3 β 1 + a 0 a 1 2 k α 3 β 2 + a 0 a 1 2 k α 3 β 3 = 0 , 2 s 2 a 1 k 3 α 3 β 1 + 1 3 a 1 3 k α 3 β 2 + 1 3 a 1 3 k α 3 β 3 = 0 .

Solving Eq. (63), we obtain the following values of constants:

(64) a 0 = a 0 , a 1 = a 1 , s = r a 1 2 a 0 , ω = k ( − 4 a 0 v α 1 + r 2 a 0 k 2 α 3 β 1 − 2 q r a 1 k 2 α 3 β 1 + 6 a 0 v 2 α 3 β 1 ) 2 a 0 , β 3 = − 3 r 2 k 2 β 1 − 2 a 0 2 β 2 2 a 0 2 .

By putting these values in Eq. (62) then in Eq. (5), we have the following solutions.

Type: 1 If r 2 − 4 q s > 0 and r s ≠ 0 or q r ≠ 0 , then

(65) u 1 ( x , t ) = a 0 − a 0 r + F tanh F 2 ( δ ) r e i ( v x + c t ) ,

(66) u 2 ( x , t ) = a 0 − a 0 r + F coth F 2 ( δ ) r e i ( v x + c t ) ,

(67) u 3 ( x , t ) = a 0 − a 0 ( r + F ( tanh ( F ( δ ) ) ± sech ( F ( δ ) ) ) ) r × e i ( v x + c t ) ,

(68) u 4 ( x , t ) = a 0 − a 0 ( r + F ( coth ( F ( δ ) ) ± csch ( F ( δ ) ) ) ) r × e i ( v x + c t ) ,

(69) u 5 ( x , t ) = a 0 + a 0 2 r + F tanh F 4 ( δ ) ± coth F 4 ( δ ) 2 r × e i ( v x + c t ) ,

(70) u 6 ( x , t ) = a 0 + a 0 − r + ( C 2 + D 2 ) ( F ) − C F cosh ( F ( δ ) ) C sinh ( F ( δ ) ) + D r × e i ( v x + c t ) ,

(71) u 7 ( x , t ) = a 0 + a 0 − r − ( C 2 + D 2 ) ( F ) + C F cosh ( F ( δ ) ) C sinh ( F ( δ ) ) + D r × e i ( v x + c t ) ,

(72) u 8 ( x , t ) = a 0 + r cosh ( F ( δ ) ) a 1 2 a 0 F sinh F 2 ( δ ) − r cosh F 2 ( δ ) × e i ( v x + c t ) ,

(73) u 9 ( x , t ) = a 0 − r sinh ( F ( δ ) ) a 1 2 a 0 r sinh F 2 ( δ ) − F cosh F 2 ( δ ) × e i ( v x + c t ) ,

(74) u 10 ( x , t ) = a 0 + r cosh F 2 ( δ ) a 1 2 a 0 ( F sinh ( F ( δ ) ) − r cosh ( F ( δ ) ) ± i F ) × e i ( v x + c t ) ,

(75) u 11 ( x , t ) = a 0 + r sinh F 2 ( δ ) a 1 2 a 0 ( F cosh ( F ( δ ) ) − r sinh ( F ( δ ) ) ± i F ) × e i ( v x + c t ) ,

(76) u 12 ( x , t ) = a 0 + 2 r sinh F 4 ( δ ) cosh F 4 ( δ ) a 1 2 a 0 ( 2 F cosh 2 F 4 − 2 r sinh F 4 cosh F 4 − F ) × e i ( v x + c t ) .

where r 2 − 2 q r a 1 a 0 = F .

Type: 2 For r 2 − 4 q s < 0 and r s ≠ 0 or q r ≠ 0 , then

(77) u 13 ( x , t ) = a 0 + a 0 − r + M tan M 2 ( δ ) r e i ( v x + c t ) ,

(78) u 14 ( x , t ) = a 0 − a 0 r + M cot M 2 ( δ ) r e i ( v x + c t ) ,

(79) u 15 ( x , t ) = a 0 + a 0 ( − r + M ( tan ( M ( δ ) ) ± sec ( M ( δ ) ) ) ) r × e i ( v x + c t ) ,

(80) u 16 ( x , t ) = a 0 − a 0 ( r + M ( cot ( M ( δ ) ) ± csc ( M ( δ ) ) ) ) r × e i ( v x + c t ) ,

(81) u 17 ( x , t ) = a 0 + a 0 − 2 r + M tan M 4 ( δ ) − cot M 4 ( δ ) 2 r × e i ( v x + c t ) ,

(82) u 18 ( x , t ) = a 0 + a 0 − r + ( C 2 + D 2 ) ( M ) − C M cos ( M ( δ ) ) C sin ( M ( δ ) ) + D r × e i ( v x + c t ) ,

(83) u 19 ( x , t ) = a 0 + a 0 − r − ( C 2 + D 2 ) ( M ) + C M cos ( M ( δ ) ) C sin ( M ( δ ) ) + D r × e i ( v x + c t ) ,

(84) u 20 ( x , t ) = a 0 − r cos ( M ( δ ) ) a 1 2 a 0 q cos M 2 ( δ ) + M sin M 2 ( δ ) × e i ( v x + c t ) ,

(85) u 21 ( x , t ) = a 0 + r sin ( M ( δ ) ) a 1 2 a 0 − q sin M 2 ( δ ) + M cos M 2 ( δ ) × e i ( v x + c t ) ,

(86) u 22 ( x , t ) = a 0 − r cos M 2 ( δ ) a 1 2 a 0 ( M sin ( M ( δ ) ) + q cos ( M ( δ ) ) ± i M ) × e i ( v x + c t ) ,

(87) u 23 ( x , t ) = a 0 + r sin M 2 ( δ ) a 1 2 a 0 ( M cos ( M ( δ ) ) − r sin ( M ( δ ) ) ± i M ) × e i ( v x + c t ) ,

(88) u 24 ( x , t ) = a 0 + 2 r sin ( M 4 ( δ ) ) cos ( M 4 ( δ ) ) a 1 2 a 0 2 M cos 2 ( M 4 ) − 2 r sin ( M 4 ) cos ( M 4 ) − M × e i ( v x + c t ) ,

where 2 q r a 1 a 0 − r 2 = M .

6 Results and discussion

In this study, we employed three different methods for various soliton solutions as follows: SEM provides multi-soliton solutions, a feature not offered by the other methods; the 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method yields bright soliton solutions, which are distinct from those generated by the other techniques; and GREMM constructs periodic singular, combined singular dark, and combined dark-bright soliton solutions, expanding the scope of soliton types investigated in our study. The advantages and limitations of these methods are given in Table 1.

Table 1

Comparison between three methods

Method	Advantages	Limitation
SEM	This method is utilized to extract multi-soliton solutions without using the Hirota bilinear form	As this method employs a specialized form of the (1+1)D Burger’s equation, it is capable of solving only (1+1)D NLPDEs to extract multiple solitons
1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method	This method yields solutions in hyperbolic form with parameter values that result in a large number of solutions, including bright, dark, and singular solitons	This method has limitation in extracting periodic-singular solutions
GREMM	GREMM constructs 27 different solutions in the form of dark, singular, periodic singular, combined dark-bright and combined dark-singular soliton solutions	GREMM fails to construct bright soliton

The outcomes obtained from the examination of the (1+1)-dimensional perturbed nonlinear Schrödinger equation are visually presented to illustrate the practical implications of the results. This section provides three-dimensional (3D), two-dimensional (2D), and contour diagrams that highlight specific findings. Both the 3D and 2D profiles of the results concerning the (1+1)-dimensional perturbed nonlinear Schrödinger equation are graphically represented. We have included a comparison of α 3 in some graphs, using 2D plots to illustrate its influence on soliton dynamics and parameter dependencies. This comparison demonstrates that the graphs shift left or right, highlighting the effect of α 3 on the positional dynamics of the solitons. Our analysis of the (1+1)-dimensional perturbed nonlinear Schrödinger equation reveals a variety of solitary wave solutions with various free parameters. These derived solutions are depicted in the graphical representations of selected results (Figures 1, 2, 3, 4, 5, 6, 7, 8, 9). We provide explanations elucidating the essence and characteristics depicted in these profiles. The results indicate that the employed rational approaches are straightforward, effective and serve as a more potent instrument. This work holds significant benefits for researchers in the fields of physics, mathematics, sciences, and engineering, opening new avenues in these disciplines. Figure 1 illustrates the graphical representation of multi-soliton in which Figure 1a shows 1-soliton solution for u 1,1 ( x , t ) with k 1 = 0.3 , ω 1 = − 40.66 , a 0 = − 1 , β 1 = 2 , β 2 = − 2 , β 3 = − 4 , α 1 = 0.1 , α 3 = − 1 , v = 1 , and c = 2 is illustrated. The 2-soliton solution for u 1,2 ( x , t ) with k 1 = 0.3 , k 2 = 0.4 , ω 1 = 2.8429 , ω 2 = 2.1222 , a 0 = − 1 , β 1 = 0.3 , β 2 = 0.1 , β 3 = − 4 , α 1 = 0.1 , α 3 = 0.11 , v = 1 , and c = 2 is depicted in Figure 1b. Furthermore, the 3-soliton solution for u 1,3 ( x , t ) with k 1 = 0.3 , k 2 = 0.35 , k 3 = 0.5 , ω 1 = 2.501 , ω 2 = 2.11307 , ω 3 = 1.395 , a 0 = − 1 , β 1 = 0.3 , β 2 = 0.1 , β 3 = − 4 , α 1 = 0.23 , α 3 = 0.1 , v = 1 , and c = 2 is presented in Figure 1c. Figure 2 shows the bright soliton solutions for the u 2,1 ( x , t ) with the parameters of k = − 2 , ω = − 2 , v = − 2 , c = 0.3 , b = 0.3 , q 0 = 1 , and q 1 = 2 . These types correspond to the lowest-amplitude dip solutions and find applications in fields such as nonlinear spectroscopy, optical communication, and ultracold atomic systems. In Figure 3, the dark soliton solutions are depicted for u 2,3 ( x , t ) with k = 2 , ω = 3 , v = 2 , c = 0.3 , and s 1 = 1 . These types of lowest amplitude at dip and finds applications in fields such as nonlinear spectroscopy optical communication, and ultracold atomic systems. Figure 4 also shows the dark soliton solution for u 1 ( x , t ) with k = 1 , ω = − 2 , v = − 2 , c = 0.5 , q = 2 , r = − 0.4 , a 0 = 0.1 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 and a 1 = 0.2 . Singular soliton is depicted in Figure 5 for u 2 ( x , t ) with k = 1 , ω = − 2 , v = − 2 , c = 0.5 , q = 2 , r = − 0.4 , a 0 = 0.1 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , and a 1 = 0.2 . These types of solitons have singular peak within wave and they help to control and manipulate wave propagation at small scale. Figure 6 depicts the combined dark-bright soliton solutions for u 3 ( x , t ) with k = 1 , ω = − 2 , v = 1 , c = 0.5 , q = 2 , r = − 1 . a 0 = 9 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , and a 1 = 0.2 . These solitons have alternating regions of low and high intensity and have applications in nonlinear fields. Figure 7 indicates the singular soliton solution of u 4 ( x , t ) with k = 1 , ω = − 2 , v = − 2 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , c = 0.5 , q = 2 , r = − 0.4 , a 0 = 0.1 , and a 1 = 0.2 . The dark-singular soliton having characteristics of both dark and singular soliton is depicted in Figure 8 for the u 5 ( x , t ) with k = 1 , ω = 0.16 , v = 2 , c = 0.003 , q = 1 , r = 0.9 , α 1 = 0.05 , α 3 = 0.05 , β 1 = 0.3 , a 0 = 2 , and a 1 = − 1 . Figure 9 indicates the behaviour of periodic singular soliton solution for u 17 ( x , t ) with k = 1 , ω = 0.016 , v = 2 , c = 0.5 , q = 2 , r = − 0.4 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , a 0 = − 1 , and a 1 = 0.2 . These solitons have singularity in their peak and regulate after specific periods and find applications in fields such as pattern formation, where their ability to generate regular wave patterns is essential for many technological applications.

Figure 1

3D plot for 1-soliton with k 1 = 0.3 , ω 1 = − 40.66 , a 0 = − 1 , β 1 = 2 , β 2 = − 2 , β 3 = − 4 , α 1 = 0.1 , α 3 = − 1 , v = 1 , and c = 2 . 3D plot for 2-soliton with k 1 = 0.3 , k 2 = 0.4 , ω 1 = 2.8429 , ω 2 = 2.1222 , a 0 = − 1 , β 1 = 0.3 , β 2 = 0.1 , β 3 = − 4 , α 1 = 0.1 , α 3 = 0.11 , v = 1 , and c = 2 . 3D plot for 3-soliton with k 1 = 0.3 , k 2 = 0.35 , k 3 = 0.5 , ω 1 = 2.501 , ω 2 = 2.11307 , ω 3 = 1.395 , a 0 = − 1 , β 1 = 0.3 , β 2 = 0.1 , β 3 = − 4 , α 1 = 0.23 , α 3 = 0.1 , v = 1 , and c = 2 . (a) 1-soliton, (b) 2-soliton, and (c) 3-soliton.

$Figure 2 3D, contour, and 2D plot for u 2,1 ( x , t ) {u}_{\mathrm{2,1}}\left(x,t) with k = − 2 k=-2 , ω = − 2 \omega =-2 , v = − 2 v=-2 , c = 0.3 c=0.3 , b = 0.3 b=0.3 , q 0 = 1 {q}_{0}=1 , and q 1 = 2 {q}_{1}=2 .$

Figure 2

3D, contour, and 2D plot for u 2,1 ( x , t ) with k = − 2 , ω = − 2 , v = − 2 , c = 0.3 , b = 0.3 , q 0 = 1 , and q 1 = 2 .

$Figure 3 3D, contour, and 2D plot for u 2,3 ( x , t ) {u}_{\mathrm{2,3}}\left(x,t) with k = 2 k=2 , ω = 3 \omega =3 , v = 2 v=2 , c = 0.3 c=0.3 , and s 1 = 1 {s}_{1}=1 .$

Figure 3

3D, contour, and 2D plot for u 2,3 ( x , t ) with k = 2 , ω = 3 , v = 2 , c = 0.3 , and s 1 = 1 .

$Figure 4 3D, contour, and 2D plot for u 1 ( x , t ) {u}_{1}\left(x,t) with k = 1 k=1 , ω = − 2 \omega =-2 , v = − 2 v=-2 , c = 0.5 c=0.5 , q = 2 q=2 , r = − 0.4 r=-0.4 , a 0 = 0.1 {a}_{0}=0.1 , α 1 = 0.1 {\alpha }_{1}=0.1 , α 3 = 0.1 {\alpha }_{3}=0.1 , β 1 = − 2 {\beta }_{1}=-2 , and a 1 = 0.2 {a}_{1}=0.2 .$

Figure 4

3D, contour, and 2D plot for u 1 ( x , t ) with k = 1 , ω = − 2 , v = − 2 , c = 0.5 , q = 2 , r = − 0.4 , a 0 = 0.1 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , and a 1 = 0.2 .

$Figure 5 3D, contour, and 2D plot for u 2 ( x , t ) {u}_{2}\left(x,t) with k = 1 k=1 , ω = − 2 \omega =-2 , v = − 2 v=-2 , c = 0.5 c=0.5 , q = 2 q=2 , r = − 0.4 r=-0.4 , a 0 = 0.1 {a}_{0}=0.1 , α 1 = 0.1 {\alpha }_{1}=0.1 , α 3 = 0.1 {\alpha }_{3}=0.1 , β 1 = − 2 {\beta }_{1}=-2 , and a 1 = 0.2 {a}_{1}=0.2 .$

Figure 5

3D, contour, and 2D plot for u 2 ( x , t ) with k = 1 , ω = − 2 , v = − 2 , c = 0.5 , q = 2 , r = − 0.4 , a 0 = 0.1 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , and a 1 = 0.2 .

$Figure 6 3D, contour, and 2D plot for u 3 ( x , t ) {u}_{3}\left(x,t) with k = 1 k=1 , ω = − 2 \omega =-2 , v = 1 v=1 , c = 0.5 c=0.5 , q = 2 q=2 , r = − 1 r=-1 . α 1 = 0.1 {\alpha }_{1}=0.1 , α 3 = 0.1 {\alpha }_{3}=0.1 , β 1 = − 2 {\beta }_{1}=-2 , a 0 = 9 {a}_{0}=9 , and a 1 = 0.2 {a}_{1}=0.2 .$

Figure 6

3D, contour, and 2D plot for u 3 ( x , t ) with k = 1 , ω = − 2 , v = 1 , c = 0.5 , q = 2 , r = − 1 . α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , a 0 = 9 , and a 1 = 0.2 .

$Figure 7 3D, contour, and 2D plot for u 4 ( x , t ) {u}_{4}\left(x,t) with k = 1 k=1 , ω = − 2 \omega =-2 , v = − 2 v=-2 , c = 0.5 c=0.5 , q = 2 q=2 , r = − 0.4 r=-0.4 , a 0 = 0.1 {a}_{0}=0.1 , and a 1 = 0.2 {a}_{1}=0.2 .$

Figure 7

3D, contour, and 2D plot for u 4 ( x , t ) with k = 1 , ω = − 2 , v = − 2 , c = 0.5 , q = 2 , r = − 0.4 , a 0 = 0.1 , and a 1 = 0.2 .

$Figure 8 3D, contour, and 2D plot for u 5 ( x , t ) {u}_{5}\left(x,t) with k = 1 k=1 , ω = 0.16 \omega =0.16 , v = 2 v=2 , c = 0.003 c=0.003 , q = 1 q=1 , r = 0.9 r=0.9 , α 1 = 0.05 {\alpha }_{1}=0.05 , α 3 = 0.05 {\alpha }_{3}=0.05 , β 1 = 0.3 {\beta }_{1}=0.3 , a 0 = 2 {a}_{0}=2 , and a 1 = − 1 {a}_{1}=-1 .$

Figure 8

3D, contour, and 2D plot for u 5 ( x , t ) with k = 1 , ω = 0.16 , v = 2 , c = 0.003 , q = 1 , r = 0.9 , α 1 = 0.05 , α 3 = 0.05 , β 1 = 0.3 , a 0 = 2 , and a 1 = − 1 .

$Figure 9 3D, contour, and 2D plot for u 17 ( x , t ) {u}_{17}\left(x,t) with k = 1 k=1 , ω = 0.016 \omega =0.016 , v = 2 v=2 , c = 0.5 c=0.5 , q = 2 q=2 , r = − 0.4 r=-0.4 , α 1 = 0.1 {\alpha }_{1}=0.1 , α 3 = 0.1 {\alpha }_{3}=0.1 , β 1 = − 2 {\beta }_{1}=-2 , a 0 = − 1 {a}_{0}=-1 , and a 1 = 0.2 {a}_{1}=0.2 .$

Figure 9

3D, contour, and 2D plot for u 17 ( x , t ) with k = 1 , ω = 0.016 , v = 2 , c = 0.5 , q = 2 , r = − 0.4 , α 1 = 0.1 , α 3 = 0.1 , β 1 = − 2 , a 0 = − 1 , and a 1 = 0.2 .

These figures portray various characteristics of soliton solutions under different parameter settings, providing valuable insights into their behaviour and dynamics. These graphical representations and analysis shed light on the intricate nature of soliton solutions in the (1+1)-dimensional perturbed nonlinear Schrödinger equation, offering valuable contributions to the understanding of nonlinear wave phenomena in optical communication systems and beyond.

7 Conclusion

In this article, we have applied the SEM, the 1 φ ( δ ) , φ ′ ( δ ) φ ( δ ) method, and the GREMM to successfully derive bright, dark, singular, combo bright-dark, periodic singular, and multi-solitons, all of which are crucial features in optical communication applications. Furthermore, we have discussed the parametric conditions necessary for the existence of optical soliton solutions.

The 2D and 3D simulations presented in this study contribute significantly to elucidating the dynamic properties of solitons arising from this model. The investigation into wave propagation in the (1+1)-dimensional perturbed nonlinear Schrödinger equation is particularly intriguing due to the presence of the perturbed term.

The observed outcomes indicate that the applied rational approaches are simple yet efficient, serving as a more potent tool for obtaining multi-soliton solutions by solving NLPDEs. This work holds considerable benefits for researchers in the fields of physics, mathematics, sciences, and engineering, providing new avenues for exploration in these disciplines.

In summary, our findings demonstrate the power and robustness of the employed analytical methods in tackling complex nonlinear problems. By shedding light on the behaviour of soliton solutions in optical communication systems, this study contributes to the advancement of our understanding of nonlinear wave phenomena and holds promise for future research in this field. In future, current study is extended by exploring the fractional and stochastic versions of this equation, which could provide deeper insights into complex wave phenomena in real-world systems.

Acknowledgments

All authors express their sincere appreciation to their respective institutions for their invaluable support. H.U.R and K.K. are grateful to HEC for the assistance provided throughout project No. 17274. Additionally, all the authors acknowledge the support of the National Science and Technology Council in Taiwan, under grant number 113-2221-E-006-033-MY3.

Funding information: All authors express their sincere appreciation to their respective institutions for their invaluable support. H.U.R and K.K. are grateful to HEC for the assistance provided throughout project No. 17274. Additionally, all the authors acknowledge the support of the National Science and Technology Council in Taiwan, under grant numbers 113-2221-E-006-033-MY3.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.
Data availability statement: All data generated or analysed during this study are included in this published article.

References

[1] Rehman HU, Awan AU, Tahir M. Singular and bright-Singular combo optical solitons in birefringent fibers to the Biswas-Arshed equation. Optik. 2020;210:164489. 10.1016/j.ijleo.2020.164489Suche in Google Scholar

[2] Rehman HU, Sultan AM, Lu D, Arshad M, Saleem MS. Soliton solutions of higher order dispersive cubic-quintic nonlinear Schrödinger equation and its applications. Chin J Phys. 2020;67:405–13. 10.1016/j.cjph.2019.10.003Suche in Google Scholar

[3] Rehman HU, Younis M, Iftikhar M. Exact solutions to the nonlinear Schrödinger and Eckhaus equations by modified simple equation method. J Adv Phys. 2014;3(1):77–9. 10.1166/jap.2014.1104Suche in Google Scholar

[4] Rehman HU, Iqbal I, Hashemi MS, Mirzazadeh M, Eslami M. Analysis of cubic-quartic-nonlinear Schrödinger equation with cubic-quintic-septic-nonic form of self-phase modulation through different techniques. Optik. 2023;287:171028. 10.1016/j.ijleo.2023.171028Suche in Google Scholar

[5] Rehman HU, Ullah N, Imran MA. Optical solitons of Biswas-Arshad equation in birefringent fibers using extended direct algebraic method. Optik. 2021;226:165378. 10.1016/j.ijleo.2020.165378Suche in Google Scholar

[6] Ablowitz MJ, Prinari B, Trubatch AD. Discrete and continuous nonlinear Schrödinger systems. Cambridge: Cambridge University Press; 2004. 10.1017/CBO9780511546709Suche in Google Scholar

[7] Ablowitz MJ. Nonlinear dispersive waves: asymptotic analysis and solitons. Cambridge: Cambridge University Press; 2011. 10.1017/CBO9780511998324Suche in Google Scholar

[8] Wazwaz AM. Bright and dark optical solitons for (3+1)-dimensional Schrödinger equation with cubic-quintic-septic nonlinearities. Optik. 2021;225:1–5. 10.1016/j.ijleo.2020.165752Suche in Google Scholar

[9] Hong WP. Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms. Opt Commun. 2001;194:217–23. 10.1016/S0030-4018(01)01267-6Suche in Google Scholar

[10] Ismael HF, Sulaiman TA, Osman MS. Multi-solutions with specific geometrical wave structures to a nonlinear evolution equation in the presence of the linear superposition principle. Commun Theor Phys. 2022;75(1):015001. 10.1088/1572-9494/aca0e2Suche in Google Scholar

[11] Malik S, Almusawa H, Kumar S, Wazwaz AW, Osman MS. A (2 + 1)-dimensional Kadomtsev-Petviashvili equation with competing dispersion effect: Painlevé analysis, dynamical behavior and invariant solutions. Results Phys. 2021;23:104043. 10.1016/j.rinp.2021.104043Suche in Google Scholar

[12] Xu N, Wen Q. Single element material sulfur quantum dots: nonlinear optics and ultrafast photonic applications. Opt Laser Technol. 2021;138:106858. 10.1016/j.optlastec.2020.106858Suche in Google Scholar

[13] Sun Y, Dong M, Yu M, Zhu L. 2bit nonlinear diffractive deep neural network (2bit ND2NN): A quantized nonlinear all-optical diffractive deep neural network implementation. Opt Laser Technol. 2024;177:111120. 10.1016/j.optlastec.2024.111120Suche in Google Scholar

[14] Wang S, Chen W, Liu W, Song D, Han X, Wang L, et al. Two-dimensional line defect lattice solitons in nonlinear fractional Schrödinger equation. Opt Laser Technol. 2024;175:110870. 10.1016/j.optlastec.2024.110870Suche in Google Scholar

[15] Hosseini K, Mirzazadeh M, Baleanu D, Salahshour S, Akinyemi L. Optical solitons of a high-order nonlinear Schrödinger equation involving nonlinear dispersions and Kerr effect. Opt. Quantum Electron. 2022;54(3):1–12. 10.1007/s11082-022-03522-0Suche in Google Scholar

[16] Chowdhury MA, Miah MM, Iqbal MA, Alshehri HM, Baleanu D, Osman MS. Advanced exact solutions to the nano-ionic currents equation through MTs and the soliton equation containing the RLC transmission line. Eur Phys J Plus. 2023;138(6):502. 10.1140/epjp/s13360-023-04105-ySuche in Google Scholar

[17] Ismael HF, Sulaiman TA, Nabi HR, Mahmoud W, Osman MS. Geometrical patterns of time-variable Kadomtsev-Petviashvili (I) equation that models dynamics of waves in thin films with high surface tension. Nonlinear Dyn. 2023;111(10):9457–66. 10.1007/s11071-023-08319-8Suche in Google Scholar

[18] Zhou Q, Sun Y, Triki H, Zhong Y, Zeng Z, Mirzazadeh M. Study on propagation properties of one-soliton in a multimode fiber with higher-order effects. Results Phys. 2022;41:105898. 10.1016/j.rinp.2022.105898Suche in Google Scholar

[19] Zhou Q, Xu M, Sun Y, Zhong Y, Mirzazadeh M. Generation and transformation of dark solitons, anti-dark solitons and dark double-hump solitons. Nonlinear Dyn. 2022;110(2):1747–52. 10.1007/s11071-022-07673-3Suche in Google Scholar

[20] Ding C, Zhou Q, Xu S, Triki H, Mirzazadeh M, Liu W. Nonautonomous breather and rogue wave in spinor Bose-Einstein condensates with space-time modulated potentials. Chin Phys Lett. 2023;40(4):040501. 10.1088/0256-307X/40/4/040501Suche in Google Scholar

[21] Savaissou N, Gambo B, Rezazadeh H, Bekir A, Doka SY. Optical solitons and dynamics in nonlinear media. Opt Quantum Electron. 2020;52:1–16. 10.1007/s11082-020-02412-7Suche in Google Scholar

[22] Jafari H, Tajadodi H, Baleanu D. Applications of a homogenous balance method to exact solutions of nonlinear fractional evolution equations. J Comput Nonlinear Dyn. 2014;9(2):021019–21. 10.1115/1.4025770Suche in Google Scholar

[23] Kudryashov NA. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simul. 2012;17:2248–53. 10.1016/j.cnsns.2011.10.016Suche in Google Scholar

[24] Rehman HU, Iqbal I, Aiadi SS, Mlaiki N, Saleem MS. Soliton solutions of Klein-Fock-Gordon equation using Sardar subequation method. Mathematics. 2022;10(18):3377. 10.3390/math10183377Suche in Google Scholar

[25] Chou D, Boulaaras SM, Rehman HU, Iqbal I. Probing wave dynamics in the modified fractional nonlinear Schrödinger equation: implications for ocean engineering. Opt Quantum Electron. 2024;56(2):228. 10.1007/s11082-023-05954-8Suche in Google Scholar

[26] Kengne E, Saydé M, Hamouda FB, Lakhssassi A. Traveling wave solutions of density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method. Eur Phys J Plus. 2013;128:1–10. 10.1140/epjp/i2013-13136-7Suche in Google Scholar

[27] Ullah N, Asjad MI, Awrejcewicz J, Muhammad T, Baleanu D. On soliton solutions of fractional-order nonlinear model appears in physical sciences. AIMS Mathematics. 2022;7:7420–40. 10.3934/math.2022415Suche in Google Scholar

[28] Ullah N, Asjad MI, Muhammad T, Akgül A. Analysis of power law non-linearity in solitonic solutions using extended hyperbolic function method. AIMS Mathematics. 2022;7:18603–15. 10.3934/math.20221023Suche in Google Scholar

[29] Ullah N, Asjad MI, Hussanan A, Akgül A, Alharbi WR, Algarni H, et al. Novel wave structures for two nonlinear partial differential equations arising in the nonlinear optics via Sardar subequation method. Alexandria Eng J. 2023;71:105–13. 10.1016/j.aej.2023.03.023Suche in Google Scholar

[30] Sagher AA, Asjad MI, Muhammad T. Advanced techniques for analyzing solitary waves in circular rods: a sensitivity visualization study. Opt Quantum Electron. 2024;56(10):1673. 10.1007/s11082-024-07573-3Suche in Google Scholar

[31] Majid SZ, Asjad MI, Riaz MB, Muhammad T. Analyzing chaos and superposition of lump waves with other waves in the time-fractional coupled nonlinear Schrödinger equation. PLoS One. 2024;19(8):e0304334. 10.1371/journal.pone.0304334Suche in Google Scholar PubMed PubMed Central

[32] Ashaq H, Majid SZ, Riaz MB, Asjad MI, Muhammad T. Exploring optical solitary wave solutions in the (2+1)-dimensional equation with in-depth dynamical assessment. Heliyon. 2024;10:e32826. 10.1016/j.heliyon.2024.e32826Suche in Google Scholar PubMed PubMed Central

[33] Yang X, Fan R, Li B. Soliton molecules and some novel interaction solutions to the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation. Phys Scr. 2020;95(4):045213. 10.1088/1402-4896/ab6483Suche in Google Scholar

[34] Akinyemi L, Houwe A, Abbagari S, Wazwaz AM, Alshehri HM, Osman MS. Effects of the higher-order dispersion on solitary waves and modulation instability in a monomode fiber. Optik. 2023;288:171202. 10.1016/j.ijleo.2023.171202Suche in Google Scholar

[35] Potasek MJ, Tabor M. Exact solutions for an extended nonlinear Schrödinger equation. Phys Lett A. 1991;154:449–52. 10.1016/0375-9601(91)90971-ASuche in Google Scholar

[36] Kodama Y. Optical solitons in a monomode fiber. J Stat Phys. 1985;39:597–614. 10.1007/BF01008354Suche in Google Scholar

[37] Rezazadeh H, Mirhosseini-Alizamini SM, Eslami M, Rezazadeh M, Mirzazadeh M, Abbagari S. New optical solitons of nonlinear conformable fractional Schrödinger-Hirota equation. Optik. 2018;172:545–53. 10.1016/j.ijleo.2018.06.111Suche in Google Scholar

[38] Daniel M, Latha MM. A generalized Davydov soliton model for energy transfer in alpha helical proteins. Physica A. 2001;298(3–4):351–70. 10.1016/S0378-4371(01)00263-1Suche in Google Scholar

[39] Hosseini K, Hincal E, Salahshour S, Mirzazadeh M, Dehingia K, Nath BJ. On the dynamics of soliton waves in a generalized nonlinear Schrödinger equation. Optik. 2023;272:170215. 10.1016/j.ijleo.2022.170215Suche in Google Scholar

[40] Kumari P, Gupta RK, Kumar S, Nisar KS. Doubly periodic wave structure of the modified Schrödinger equation with fractional temporal evolution. Results Phys. 2022;33:105128. 10.1016/j.rinp.2021.105128Suche in Google Scholar

[41] Zayed EM, Alngar ME. Optical solitons in birefringent fibers with Biswas-Arshed model by generalized Jacobi elliptic function expansion method. Optik. 2020;203:163922. 10.1016/j.ijleo.2019.163922Suche in Google Scholar

[42] Kuo CK. A novel method for finding new multi-soliton wave solutions of the completely integrable equations. Optik. 2017;139:283–90. 10.1016/j.ijleo.2017.04.014Suche in Google Scholar

[43] Arnous AH, Nofal TA, Biswas A, Yildirim Y, Asiri A. Cubic-quartic optical solitons of the complex Ginzburg-Landau equation: A novel approach. Nonlinear Dyn. 2023:1–16. 10.1049/ote2.12065Suche in Google Scholar

[44] Ahmed N, Baber MZ, Iqbal MS, Annum A, Ali SM, Ali M, El Din SM. Analytical study of reaction diffusion Lengyel-Epstein system by generalized Riccati equation mapping method. Sci Rep. 2023;13(1):20033. 10.1038/s41598-023-47207-4Suche in Google Scholar PubMed PubMed Central

[45] Ozisik M. On the optical soliton solution of the (1+1)-dimensional perturbed NLSE in optical nano-fibers. Optik. 2022;250:168233. 10.1016/j.ijleo.2021.168233Suche in Google Scholar

[46] Javed S, Ali A, Muhammad T. Dynamical perspective of bifurcation analysis and soliton solutions to (1+1)-dimensional nonlinear perturbed Schrödinger model. Opt. Quantum Electron. 2024;56(6):1013. 10.1007/s11082-024-06926-2Suche in Google Scholar

[47] Ullah N, Ullah A, Ali S, Ahmad S. Optical solitons solution for the perturbed nonlinear Schrödingeras equation. Partial Differ Equ Appl Math. 2024;11:100837. 10.1016/j.padiff.2024.100837Suche in Google Scholar

[48] Akbar MA, Wazwaz AM, Mahmud F, Baleanu D, Roy R, Barman HK, et al. Dynamical behavior of solitons of the perturbed nonlinear Schrödinger equation and microtubules through the generalized Kudryashov scheme. Results Phys. 2022;43:106079. 10.1016/j.rinp.2022.106079Suche in Google Scholar

Received: 2024-10-03

Revised: 2024-12-22

Accepted: 2025-06-16

Published Online: 2025-11-03

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

Artikel in diesem Heft

https://doi.org/10.1515/nleng-2025-0166

Schlagwörter für diesen Artikel

optical solitons; nonlinear Schrödinger equation; analytical methods; parametric analysis; magneto-optic waveguides

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