Insights from the nonlinear Schrödinger–Hirota equation with chromatic dispersion: Dynamics in fiber–optic communication

Dean Chou; Hamood Ur Rehman; Muhammad Imran Asjad; Ifrah Iqbal

doi:10.1515/nleng-2024-0058

Article Open Access

Insights from the nonlinear Schrödinger–Hirota equation with chromatic dispersion: Dynamics in fiber–optic communication

Dean Chou , Hamood Ur Rehman , Muhammad Imran Asjad and Ifrah Iqbal

Published/Copyright: April 29, 2025

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From the journal Nonlinear Engineering Volume 14 Issue 1

Abstract

Optical solitons have practical applications in communication systems as carriers of optical information. An advantage of an optical soliton is its ability to maintain its structure unchanged when interacting with other pulses. The purpose of this article is to strive for the optical soliton solutions of the nonlinear Schrödinger–Hirota equation incorporating chromatic dispersion, which is a governing model for understanding the dynamics of dispersive pulse propagation in optical fibers. Three distinct methodologies, namely, the 1 φ ( η ) , φ ′ ( η ) φ ( η ) method, the new Kudryashov method, and the extended simple equation method, have been employed to derive solutions for this equation. Numerous optical soliton solutions including singular soliton, dark soliton, bright soliton, periodic singular soliton, dark–bright soliton, and dark-singular solution and singular periodic solutions are offered by this method. These techniques present an accurate and successful strategy for deriving exact solutions to various nonlinear partial differential equations. Graphical representations of some of the extracted solutions are shown to demonstrate the behavior of solitons. We also presented the eye diagram for the depiction of the signal to provide a clear visualization of the pulse shapes and their characteristics. The optical solitons produced in respect to this form have never been explored by the proposed techniques before, and the results have never been published. In this study, we perform a comprehensive perturbation analysis to explore the stability and dynamics of soliton solutions within the Schrödinger–Hirota equation.

Keywords: nonlinear Schrödinger equation; optical soliton; chromatic dispersion; (1/φ(η), φ′(η)/φ(η)) method; new Kudryashov method; extended simple equation method

1 Introduction

The exploration of optical soliton solutions to nonlinear partial differential equations (NPDEs) has emerged as a captivating and pivotal area of investigation within the realms of applied sciences and engineering [1–14]. Solitons, possessing numerous applications in science and engineering play a crucial role in various technological advancements. A paramount application of solitons lies in their ability to transmit digital information through optical fibers. The utilization of solitons in conveying information has become a cornerstone in the technological landscape [15–22]. Additionally, solitons are subject to intensive examination in electromagnetics, serving as transverse electromagnetic waves passing between two strips of superconducting metal. In the field of nonlinear optics, optical solitons constitute a remarkable field of study including optical fibers, magneto-optics, optical couplers, crystals, meta-surfaces, and birefringent materials [23–25]. The detailed study of optical solitons adds depth to our understanding of nonlinear optical phenomena and contributes in various optical technologies.

Recently, a significant focus in scientific research has been directed toward exploring mathematical models that elucidate the propagation of pulses in optical media. This heightened interest is primarily attributed to the rapid advancements in information transmission systems, especially over long distances. Among the widely employed nonlinear equations for characterizing pulse propagation in optical media is the nonlinear Schrödinger-Hirota equation (NLSHE) [26–28]. The NLSE is an NPDE that has application in diverse physical scenarios incluiding fluid mechanics, hydrodynamic, and the depiction of surface gravity water wave evolution. Moreover, it serves as a fundamental model in various fields, including nonlinear optics, fluid dynamics, nuclear physics, mathematical finance, biochemistry, plasma physics, and superconductivity. Specifically, it describes phenomena such as solitary wave propagation in heat pulse propagation in solids, piezoelectric semiconductors in condensed matter physics, and more. Another variant that has garnered attention in several studies is the NLSHE. This equation accounts for third-order dispersion (TOD) and finds application in the analysis of pulse propagation processes within optical media [29].

Chromatic dispersion (CD) and nonlinear refractive index (NRI) represent essential components in the foundational principles governing soliton propagation within optical fibers [30]. CD arises due to the dependency of the phase and group velocity of light in a transparent medium on optical frequency. This dependence leads to variations in wavelength and propagation velocity [31]. While the impact of CD in fibers can be mitigated, it is an inherent phenomenon in the production of glass fibers. Managing CD becomes imperative for optimizing and minimizing the effects of nonlinear terms in wavelength-division multiplexing systems. Despite efforts to reset the effects of CD, achieving a perfect balance remains a significant challenge. CD introduces issues in fibers, manifesting as degraded signal quality, necessitating dispersion compensation. A highly effective approach to address CD concerns is the utilization of fiber gratings [32]. The NLSHE with CD is given as follows [33,34]:

(1) ι y t + a 1 y x x + b 1 ∣ y ∣ 2 y + ι ( c 1 y x x x + d 1 ∣ y ∣ 2 y x ) = 0 ,

where a 1 , b 1 , c 1 , and d 1 show the coefficients associated with group velocity dispersion (GVD), Kerr-law nonlinearity, TOD, and nonlinear dispersion terms, respectively.

The NLSHE with CD, as presented in our model, is applicable to dispersion-flattened fibers and waveguides exhibiting polarization-mode dispersion (PMD). The addition of Kerr nonlinearity, GVD, TOD, and nonlinear dispersion terms accurately investigates the behavior of optical pulses in these systems. Dispersion-flattened fibers are particularly relevant here due to the need for managing both second- and third-order dispersive effects, while the nonlinear dispersion term addresses the impact of PMD in birefringent fibers. This model can be extended to grating structures, and its primary applicability is in fiber optic communication systems where higher-order nonlinearities and dispersion significantly affects the soliton dynamics. The fibers considered in our model are step-index fibers. This choice is due to their distinct core and cladding refractive index profiles, which allow us to analyze the nonlinear dynamics and higher-order dispersive effects, such as GVD and TOD, more clearly. Step-index fibers, particularly in terms of dispersion-flattened designs, give a straightforward framework for studying soliton propagation, PMD, and other nonlinear effects, which are central to the model.

Numerous methodologies have been developed to extract soliton solutions from NPDEs, including, but not limited to, the following: Kudryashov’s method [35,36], extended hyperbolic function method [37–41], Bernoulli sub-ODE method [42,43], Sardar sub-equation method [44,45], newly extended direct algebraic method [46,47], Jacobi’s elliptic function [48,49], mapping method [50–53], and ϕ 6 -expansion method [54]. In this study, we have adopted three distinct approaches, namely, the 1 φ ( η ) , φ ′ ( η ) φ ( η ) method [21], the new Kudryashov method [55], and the extended simple equation method [56], for the first time. Through these novel approaches, we have successfully extracted diverse solutions, including dark, bright, periodic singular, singular, combined dark–bright, and combined dark-singular optical solitons.

The remaining structure of this article is as follows: Section 2 gives mathematical analysis of the equation. Sections 3, 4, and 5 give the descriptions of applied methods with applications. The perturbation analysis for slowdown effect is given in Section 6. The discussion of this article is given in Section 7. Section 8 is of conclusion.

2 Mathematical analysis

Suppose

(2) y ( x , t ) = Y ( η ) e ι ξ , η = x − v t , and ξ = − α x + β t + ψ ,

where v , α , β , and ψ represent the velocity, frequency, wave number, and phase constant, respectively. By substituting (2) into (1), the real and imaginary parts are obtained as follows:

(3) ( α d 1 + b 1 ) Y 3 + ( − α 3 c 1 − a 1 α 2 − β ) Y + ( 3 α c 1 + a 1 ) Y ′ ′ = 0 ,

(4) d 1 Y 2 Y ′ + ( 3 α 2 c 1 − 2 a 1 α − v ) Y ′ + c 1 Y ′ ′ ′ = 0 .

By integrating (4) and let integration of constant equals to zero

(5) d 1 Y 3 + ( − 9 α 2 c 1 − 6 a 1 α − 3 v ) Y + 3 c 1 Y ′ ′ = 0 .

Now, from (3) and (5), we obtained the following values of constants:

(6) d 1 = 3 b 1 c 1 a 1 , β = − 8 α 3 c 1 2 + 8 a 1 α c 1 + 2 a 1 2 α + 3 α c 1 v + a 1 v c 1 .

Under these conditions, consider (3) or (5) as nonlinear ordinary differential equation (ODE).

3 Description of 1 φ ( η ) , φ ′ ( η ) φ ( η ) method

Let the solution of (3) be

(7) Y ( η ) = w 0 + ∑ i = 1 m w i + v i φ ′ ( η ) i φ ( η ) i ,

where w 0 , w i , and v i ( i = 1 , 2 , … , m ) are the constants. m can be obtained using the homogeneous balancing rule and φ ( η ) represents the following ODE:

(8) φ ′ ( η ) 2 = φ ( η ) 2 − ϱ ,

where

(9) φ ( η ) = c e η + ϱ 4 c e η .

Now, by substituting (7) along (8) into (3), the system of equations is attained, and by solving it, we obtain the values of constants.

3.1 Application of 1 φ ( η ) , φ ′ ( η ) φ ( η ) method

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

(10) Y ( η ) = w 0 + w 1 + v 1 φ ′ ( η ) φ ( η ) .

Now, by substituting Eq. (10) along Eq. (8) and Eq. (9) into (3) , the system of equations is attained, and by solving it, we obtain the following values of constants:

Set 1:

w 0 = 0 , w 1 = v 1 − ϱ ( 6 a 2 + 4 α 2 + 15 ) 6 a 2 − 4 α 2 + 3 , c 1 = 2 ( 2 a 2 β + 3 β ) α ( 2 α 2 + 3 ) , b 1 = 4 α 2 β − 6 a 2 β − 3 β − 2 α 3 d 1 v 1 2 − 3 α d 1 v 1 2 ( 2 α 2 + 3 ) v 1 2 , a 1 = − 2 β .

By substituting these values into Eq. (10), we have

y 1 ( x , t ) = v 1 4 c e η − ϱ ( 6 a 2 + 4 α 2 + 15 ) 6 a 2 − 4 α 2 + 3 + c e η − ϱ 4 c 2 e 2 η + ϱ e ι ξ .

By taking ϱ = ± 4 c 2 , we obtain

(11) y 1,1 ( x , t ) = v 1 − ( 6 a 2 + 4 α 2 + 15 ) sech ( η ) 6 a 2 − 4 α 2 + 3 + tanh ( η ) e ι ξ .

(12) y 1,2 ( x , t ) = v 1 6 a 2 + 4 α 2 + 15 csch ( η ) 6 a 2 − 4 α 2 + 3 + coth ( η ) e ι ξ .

Set 2:

v 1 = 0 , d 1 = a 1 b 1 3 ( a 1 a 2 + β ) , α = − 3 ( a 1 a 2 + β ) a 1 , c 1 = a 1 3 ∕ 2 3 3 ( a 1 a 2 + β ) .

By substituting these values into Eq. (10), we have

y 1 , * ( x , t ) = 4 c w 1 e η 4 c 2 e 2 η + ϱ + w 0 e ι ξ .

By taking ϱ = ± 4 c 2 , we obtain

(13) y 1,3 ( x , t ) = w 1 sech ( η ) 2 c + w 0 e ι ξ ,

(14) y 1,4 ( x , t ) = w 1 csch ( η ) 2 c + w 0 e ι ξ .

Set 3:

w 0 = 0 , w 1 = 0 , c 1 = − a 1 a 2 + 2 a 1 + β α ( α 2 + 6 ) , d 1 = 6 a 1 a 2 + 2 α 2 a 1 − α 2 b 1 v 1 2 − 6 b 1 v 1 2 − 6 β ( α 3 + 6 α ) v 1 2 .

By substituting these values into Eq. (10), we have

y 1 , * * ( x , t ) = v 1 ( 4 d 2 e 2 η − ϱ ) 4 c 2 e 2 η + ϱ e ι ξ .

By taking ϱ = ± 4 c 2 , we obtain

(15) y 1,5 ( x , t ) = ( v 1 tanh ( η ) ) e ι ξ ,

(16) y 1,6 ( x , t ) = ( v 1 coth ( η ) ) e ι ξ .

4 New Kudryashov’s method

Let the solution of (3) be

(17) Y ( η ) = ∑ i = 1 M h i χ ( η ) ,

where h i ( i = 1 , 2 , … , M ) are the constants. M can be obtained by using the homogeneous balancing rule, and χ ( η ) represents the following ODE:

(18) χ ′ ( η ) 2 = δ 2 χ ( η ) 2 ( 1 − λ χ ( η ) 2 ) ,

where

(19) χ ( η ) = 4 R 4 e δ η R 2 + e − δ η λ .

Now, by substituting (17) along (18) into (3), the system of equations is attained, and by solving it, we obtain the values of constants.

4.1 Application of new Kudryashov’s method

As the balancing number M = 1 , so from (17), we have

(20) Y ( η ) = h 0 + h 1 χ ( η ) .

By substituting (20) along (18) into (3), the system of equations is attained, and by solving it, we obtain the values of constants:

h 0 = 0 , a 1 = − β + α 3 ( − c 1 ) + 3 α c 1 δ 2 a 2 − δ 2 , b 1 = − a 2 α h 1 2 d 1 + 6 a 2 α c 1 δ 2 λ − 2 β δ 2 λ + α h 1 2 δ 2 d 1 − 2 α 3 c 1 δ 2 λ h 1 2 ( a 2 − δ 2 ) .

By substituting these values of constants into (20), we have the following solutions:

(21) y 2 ( x , t ) = 4 h 1 R 4 R 2 e δ η + λ e δ ( − η ) e ι ξ .

By taking λ = ± 4 R 2 , we have the following solutions:

(22) y 2,1 ( x , t ) = h 1 sech ( δ η ) 2 R e ι ξ ,

(23) y 2,2 ( x , t ) = h 1 csch ( δ η ) 2 R e ι ξ .

5 Extended simple equation method

Consider the solution of (3) is

(24) Y ( η ) = ∑ i = − M M g i F ( η ) ,

where g i ( i = 1 , 2 , … , M ) are the constants. M can be obtained by using the homogeneous balancing rule, and F ( η ) represents the following ODE:

(25) F ′ ( η ) = p 3 F ( η ) 3 + p 2 F ( η ) 2 + p 1 F ( η ) + p 0 ,

where p i , i = 0 , 1, 2, 3 are the arbitrary constants and (25) have the following solutions:

Case 1 When p 0 = p 3 = 0 ,

F 1 ( η ) = p 1 exp ( η p 1 ) 1 − p 2 exp ( η p 1 ) , p 1 > 0 , F 2 ( η ) = − p 1 exp ( η p 1 ) 1 − p 2 exp ( η p 1 ) , p 1 < 0 .

Case 2 When p 1 = p 3 = 0 ,

F 3 ( η ) = p 0 p 2 tan ( p 0 p 2 η ) p 2 , p 0 p 2 > 0 , F 4 ( η ) = − p 0 p 2 tanh ( − p 0 p 2 η ) p 2 , p 0 p 2 < 0 .

Case 3 When p 0 = p 2 = 0 ,

F 5 ( η ) = p 1 exp ( η p 1 ) 1 − exp ( 2 η p 1 p 3 ) , p 1 > 0 , F 6 ( η ) = − p 1 exp ( η p 1 ) exp ( − 2 η p 1 p 3 ) , p 1 < 0 .

5.1 Application of the extended simple equation method

As homogeneous balancing rule gives M = 1 , so we acquired from (24):

(26) Y ( η ) = g 0 + g 1 F ( η ) + g − 1 F ( η ) − 1 .

By substituting (26) and (25) into (3), we obtain following sets of solutions:

Set 1 a 1 = − 3 α c 1 , b 1 = − α d 1 , β = − α c 1 ( α 2 − 3 a 2 ) .

Using these values along solutions of (25), we obtained

Case 1 When p 0 = p 3 = 0 ,

y 3,1 ( η ) = g − 1 ( e − η p 1 − p 2 ) p 1 + g 1 p 1 e η p 1 1 − p 2 e η p 1 + g 0 e ι ξ , y 3,2 ( η ) = g − 1 ( − e η p 1 + p 2 ) p 1 + g 1 p 1 e η p 1 − 1 + p 2 e η p 1 + g 0 e ι ξ .

Case 2 When p 1 = p 3 = 0 ,

y 3,3 ( η ) = g 0 − g − 1 p 2 coth ( η − p 0 p 2 ) + g 1 p 0 tanh ( η − p 0 p 2 ) − p 0 p 2 e ι ξ , y 3,4 ( η ) = g 0 + g − 1 p 2 cot ( η p 0 p 2 ) + g 1 p 0 tan ( η p 0 p 2 ) p 0 p 2 e ι ξ .

Case 3 When p 0 = p 2 = 0 ,

y 3,5 ( η ) = g − 1 1 − e 2 η p 1 p 3 p 1 e η p 1 + g 1 p 1 e η p 1 1 − e 2 η p 1 p 3 + g 0 e ι ξ , y 3,6 ( η ) = g − 1 e − 2 η p 1 p 3 p 1 ( − e η p 1 ) + g 1 p 1 ( − e η p 1 ) e − 2 η p 1 p 3 + g 0 e ι ξ .

Set 2 g 1 = 0 , b 1 = − α d 1 , a 1 = − 3 α c 1 , β = − α c 1 ( α 2 − 3 a 2 ) .

Using these values along solutions of (25), we obtained

Case 1 When p 0 = p 3 = 0 ,

y 3,7 ( η ) = g − 1 e − η p 1 − g − 1 p 2 + p 1 g 0 p 1 e ι ξ , y 3,8 ( η ) = g − 1 e η p 1 + g − 1 p 2 + p 1 g 0 p 1 e ι ξ .

Case 2 When p 1 = p 3 = 0 ,

y 3,9 ( η ) = g 0 + g − 1 p 2 cot ( η p 0 p 2 ) p 0 p 2 e ι ξ , y 3,10 ( η ) = g 0 + g − 1 p 2 coth ( η − p 0 p 2 ) − p 0 p 2 e ι ξ .

Case 3 When p 0 = p 2 = 0 ,

y 3,11 ( η ) = g − 1 1 − e 2 η p 1 p 3 p 1 e η p 1 + g 0 e ι ξ , y 3,12 ( η ) = g − 1 e − 2 η p 1 p 3 p 1 ( − e η p 1 ) + g 0 e ι ξ .

Set 3 g − 1 = 0 , b 1 = − α d 1 , a 1 = − 3 α c 1 , β = − α c 1 ( α 2 − 3 a 2 ) .

Using these values along solutions of (25), we obtained

Case 1 When p 0 = p 3 = 0 ,

y 3,13 ( x , t ) = g 0 + g 1 p 1 exp ( η p 1 ) 1 − p 2 exp ( η p 1 ) e ι ξ , y 3,14 ( x , t ) = g 0 + g 1 − p 1 exp ( η p 1 ) 1 − p 2 exp ( η p 1 ) e ι ξ .

Case 2 When p 1 = p 3 = 0 ,

y 3,15 ( x , t ) = g 0 + g 1 p 0 p 2 tan ( p 0 p 2 η ) p 2 e ι ξ , y 3,16 ( x , t ) = g 0 + g 1 − p 0 p 2 tanh ( − p 0 p 2 η ) p 2 e ι ξ .

Case 3 When p 0 = p 2 = 0 ,

y 3,17 ( x , t ) = g 0 + g 1 p 1 exp ( η p 1 ) 1 − exp ( 2 η p 1 p 3 ) e ι ξ , y 3,18 ( x , t ) = g 0 + g 1 − p 1 exp ( η p 1 ) exp ( − 2 η p 1 p 3 ) e ι ξ .

6 Perturbative analysis

In this section, we extend our analysis of the soliton solution by incorporating the effects of TOD. The presence of TOD is known to induce both a slowdown in the soliton’s velocity and the emission of radiation in the form of small-amplitude dispersive waves. To obtain these physical effects, the soliton solution is perturbed by adding a small parameter, ε , which accounts for the deviations caused by TOD [57].

We begin with the unperturbed soliton solution, which is given by

(27) y 1,1 ( x , t ) = h 1 csch ( δ η ) 2 R e i ξ , y 2,1 ( x , t ) = h 1 sech ( δ η ) 2 R e i ξ ,

where η = x − v t shows the soliton’s core moving at velocity v , h 1 presents the soliton amplitude, and R is a scaling parameter. The phase factor ξ = α x − β t describes the wave number α and frequency β of the soliton. The sech ( δ η ) proposes the soliton’s localized shape, with δ controlling the soliton width.

To incorporate the effects of TOD, a small perturbation parameter ε is introduced to modify both the soliton’s velocity (slowdown effect) and induced radiation. The perturbed soliton solution is written as:

(28) y 1,1 ( x , t ) = h 1 csch ( δ η slowdown ) 2 R e i ξ slowdown + ε A e − γ t e i ( κ x − ω r t ) , y 2,1 ( x , t ) = h 1 sech ( δ η slowdown ) 2 R e i ξ slowdown + ε A e − γ t e i ( κ x − ω r t ) ,

where η slowdown = x − ( v + ε t ) t introduces a time-dependent velocity correction, representing the soliton slowdown due to TOD. Similarly, the phase factor is modified as:

(29) ξ slowdown = α x − ( β + ε t ) t .

The second term in (28), ε A e − γ t e i ( κ x − ω r t ) , represents the radiative component, where A is the amplitude of the emitted radiation, γ is a decay constant that controls how quickly the radiation diminishes over time, κ is the wavenumber of the radiative wave, and ω r is the frequency of the radiative wave. This term models the small-amplitude dispersive waves that are emitted by the soliton as it propagates and gradually loses energy due to the perturbation. The introduction of the perturbation term has two key effects: the soliton experiences a time-dependent reduction in its velocity due to TOD. This is formed by the perturbative shift in the soliton’s velocity and is given by ( v + ε t ) , where ε shows the magnitude of the slowdown. As time progresses, the soliton decelerates, as evidenced by the shift in the position of the soliton core. The perturbation also triggers the emission of radiation in the form of small dispersive waves. These waves, described by the oscillatory term ε A e − γ t e i ( κ x − ω r t ) , gradually radiate energy away from the soliton. The amplitude of these radiative waves decays over time due to the factor e − γ t , and their wavenumber κ and frequency ω r characterize their oscillatory behavior. To visualize the effects of the perturbation, we numerically solved (28) for both the unperturbed and perturbed soliton solutions. The results are shown in Figure 8.

$Figure 1 Plots of y 1,1 ( x , t ) {y}_{\mathrm{1,1}}\left(x,t) by taking α = 1.5 \alpha =1.5 , a = 0.5 , β = 1.3 a=0.5,\beta =1.3 , v 1 = 0.5 {v}_{1}=0.5 , v = 1.5 v=1.5 and ψ = 0.5 \psi =0.5 .$

Figure 1

Plots of y 1,1 ( x , t ) by taking α = 1.5 , a = 0.5 , β = 1.3 , v 1 = 0.5 , v = 1.5 and ψ = 0.5 .

In the study by Kaur et al. Figure 8, we observe that the perturbed soliton (red dashed line) shifts to the left compared to the original soliton (blue solid line), indicating the deceleration caused by the perturbation. Furthermore, small oscillations trailing the soliton illustrate the radiative waves that are emitted due to the perturbation.

7 Results and discussion

In the study by Kaur et al. [58], the Schrödinger–Hirota equation with variable coefficients and power-law nonlinearity was investigated, focusing on bright and dark optical solitons. Similarly, bright, dark, and singular soliton solutions of the perturbed Schrödinger–Hirota equation, incorporating spatio-temporal dispersion and Kerr nonlinearity, were developed in the study by Yildirim [59]. New soliton solutions of the time-fractional Schrödinger–Hirota equation were explored in the study by Zayed et al. [60], where various dispersive optical soliton solutions were derived. The dynamic behaviors and optical solitons within DWDM networks, modeled by the Schrödinger–Hirota equation, were examined in the study by Tang [61] using traveling wave reductions to a plane dynamical system. The existing literature, as referenced, establishes that while numerous aspects of the Schrödinger–Hirota equation have been explored, our work presents a novel contribution, further advancing the field by addressing gaps in previous studies and introducing new solutions within this framework.

This study delves into the exploration of soliton solutions for the NLSHE through the application of three distinct approaches. The obtained solutions encompass various types, providing a comprehensive understanding of the equation’s behavior. In this section, we present graphical representations of these solutions through 3D and 2D plots, showcasing absolute, real, and imaginary aspects. These visualizations serve as crucial tools for comprehending the intricate dynamics of waves and play a vital role in validating the accuracy of the obtained solutions. The eye diagram is presented for the signal, providing a clear visualization of the pulse shapes and their characteristics. The graphical representations delineate diverse wave patterns influenced by constants, with potential applications in marine engineering for the design of resilient structures and the control of wave propagation. The details of each figure are elucidated as follows:

Figure 1(a)–(c) presents the absolute value graph of y 1,1 ( x , t ) , revealing a distinctive combined dark–bright optical soliton. In Figure 1(d)–(f), the real value graphs of y 1,1 ( x , t ) illustrate the periodic patterns. Figure 1(g)–(i) portrays the imaginary value graphs of y 1,1 ( x , t ) , exhibiting periodic characteristics similar to the real value graphs but with differences in amplitude. These graphical representations correspond to the specified constants: α = 1.5 , a = 0.5 , β = 1.3 , v 1 = 0.5 , v = 1.5 and ψ = 0.5 . Figure 2(a)–(c) showcases the absolute value graph of y 1,4 ( x , t ) , unveiling a distinctive singular optical soliton.

$Figure 2 Plots of y 1,4 ( x , t ) {y}_{\mathrm{1,4}}\left(x,t) by taking α = 1.5 \alpha =1.5 , a = 0.5 , β = 1.3 a=0.5,\beta =1.3 , w 1 = 0.5 {w}_{1}=0.5 , v = 1.5 v=1.5 , w 0 = 0.5 {w}_{0}=0.5 , c = 1 c=1 , and ψ = 0.5 \psi =0.5 .$

Figure 2

Plots of y 1,4 ( x , t ) by taking α = 1.5 , a = 0.5 , β = 1.3 , w 1 = 0.5 , v = 1.5 , w 0 = 0.5 , c = 1 , and ψ = 0.5 .

$Figure 3 Plots of y 1,5 ( x , t ) {y}_{\mathrm{1,5}}\left(x,t) by taking α = 1.5 \alpha =1.5 , a = 0.5 , β = 1.3 a=0.5,\beta =1.3 , v = 1.5 v=1.5 , v 1 = 1.5 {v}_{1}=1.5 , and ψ = 0.5 \psi =0.5 .$

Figure 3

Plots of y 1,5 ( x , t ) by taking α = 1.5 , a = 0.5 , β = 1.3 , v = 1.5 , v 1 = 1.5 , and ψ = 0.5 .

In Figure 2(d)–(f), the real value graphs of y 1,4 ( x , t ) illustrate the periodic-singular patterns. Figure 2(g)–(i) depict the imaginary value graphs of y 1,4 ( x , t ) , revealing periodic singular characteristics akin to the real value graphs but with variations in amplitude. These graphical representations correspond to the specified constants: α = 1.5 , a = 0.5 , β = 1.3 , w 1 = 0.5 , v = 1.5 , w 0 = 0.5 , c = 1 , and ψ = 0.5 .

Figure 3(a)–(c) showcases the absolute value graph of y 1,5 ( x , t ) , unveiling a distinctive dark optical soliton. In Figures 3(d)–(f), the real value graphs of y 1,5 ( x , t ) illustrate the periodic patterns. Figure 3(g)–(i) depicts the imaginary value graphs of y 1,5 ( x , t ) , revealing periodic characteristics same to the real value graphs but with variations in amplitude. These graphical representations correspond to the specified constants: α = 1.5 , a = 0.5 , β = 1.3 , v = 1.5 , v 1 = 1.5 , and ψ = 0.5 .

$Figure 4 Plots of y 2,1 ( x , t ) {y}_{\mathrm{2,1}}\left(x,t) by taking α = 1.5 \alpha =1.5 , a = 0.5 , β = 1.3 a=0.5,\beta =1.3 , v = 1.5 v=1.5 , R = 1 R=1 , h 1 = 1 {h}_{1}=1 , δ = 0.45 \delta =0.45 , and ψ = 0.5 \psi =0.5 .$

Figure 4

Plots of y 2,1 ( x , t ) by taking α = 1.5 , a = 0.5 , β = 1.3 , v = 1.5 , R = 1 , h 1 = 1 , δ = 0.45 , and ψ = 0.5 .

Figure 4(a)–(c) showcases the absolute value graph of y 2,1 ( x , t ) , which represents the bright optical soliton. In Figure 4(d)–(f), the real value graphs of y 2,1 ( x , t ) illustrate the dark–bright optical soliton. Figure 4(g)–(i) depicts the imaginary value graphs of y 2,1 ( x , t ) , revealing dark–bright characteristics same to the real value graphs but with variations in amplitude. These graphical representations correspond to the specified constants: α = 1.5 , a = 0.5 , β = 1.3 , v = 1.5 , R = 1 , h 1 = 1 , δ = 0.45 , and ψ = 0.5 .

$Figure 5 Plots of y 3,3 ( x , t ) {y}_{\mathrm{3,3}}\left(x,t) by taking α = 1.5 \alpha =1.5 , g 1 = 2 {g}_{1}=2 , a = 0.5 , β = 1.3 a=0.5,\beta =1.3 , v = 1.5 v=1.5 , p 0 = 0.1 , p 2 = − 0.2 {p}_{0}=0.1,{p}_{2}=-0.2 , g − 1 = 1 , g 0 = 0.5 {g}_{-1}=1,{g}_{0}=0.5 , and ψ = 0.5 \psi =0.5 .$

Figure 5

Plots of y 3,3 ( x , t ) by taking α = 1.5 , g 1 = 2 , a = 0.5 , β = 1.3 , v = 1.5 , p 0 = 0.1 , p 2 = − 0.2 , g − 1 = 1 , g 0 = 0.5 , and ψ = 0.5 .

Figure 5(a)–(c) showcases the absolute value graph of y 3,3 ( x , t ) , which represents the combined dark–singular optical soliton. In Figure 5(d)–(f), the real value graphs of y 3,3 ( x , t ) illustrate the singular optical soliton. Figure 5(g)–(i) depicts the imaginary value graphs of y 3,3 ( x , t ) , revealing singular optical solitons similar to the real value graphs but with variations in amplitude. These graphical representations correspond to the specified constants: α = 1.5 , a = 0.5 , β = 1.3 , v = 1.5 , p 0 = 0.1 , p 2 = − 0.2 , g − 1 = 1 , g 0 = 0.5 , and ψ = 0.5 .

$Figure 6 Plots of y 3,5 ( x , t ) {y}_{\mathrm{3,5}}\left(x,t) by taking α = 0.1 \alpha =0.1 , a = 0.99 a=0.99 , β = 1 , g − 1 = − 0.5 , g 1 = 0.5 , p 1 = 0.1 , p 3 = − 0.2 , v = 0.5 \beta =1,{g}_{-1}=-0.5,{g}_{1}=0.5,{p}_{1}=0.1,{p}_{3}=-0.2,v=0.5 , and g 0 = 1 . {g}_{0}=1. .$

Figure 6

Plots of y 3,5 ( x , t ) by taking α = 0.1 , a = 0.99 , β = 1 , g − 1 = − 0.5 , g 1 = 0.5 , p 1 = 0.1 , p 3 = − 0.2 , v = 0.5 , and g 0 = 1 . .

Figure 6(a)–(c) depicts the absolute value graph of y 3,5 ( x , t ) , which represents the singular optical soliton. In Figure 6(d)–(f), the real value graphs of y 3,5 ( x , t ) illustrate the singular optical soliton. Figures 6(g)–(i) depicts the imaginary value graphs of y 3,5 ( x , t ) , revealing singular optical solitons same to the real value graphs but with variations in amplitude under specified constants: α = 0.1 , a = 0.99 , β = 1 , g − 1 = − 0.5 , g 1 = 0.5 , p 1 = 0.1 , p 3 = − 0.2 , v = 0.5 , and g 0 = 1 .

$Figure 7 Plots of y 3,15 ( x , t ) {y}_{\mathrm{3,15}}\left(x,t) by taking α = 0.1 \alpha =0.1 , a = 0.99 a=0.99 , β = 1 , g 1 = − 0.5 , p 0 = 0.1 , p 2 = 0.2 , v = 0.5 \beta =1,{g}_{1}=-0.5,{p}_{0}=0.1,{p}_{2}=0.2,v=0.5 , and g 0 = 1 . {g}_{0}=1.$

Figure 7

Plots of y 3,15 ( x , t ) by taking α = 0.1 , a = 0.99 , β = 1 , g 1 = − 0.5 , p 0 = 0.1 , p 2 = 0.2 , v = 0.5 , and g 0 = 1 .

Figure 7(a)–(c) illustrates the absolute value graph of y 3,15 ( x , t ) , which represents the periodic-singular optical soliton. In Figure 7(d)–(f), the real value graphs of y 3,15 ( x , t ) illustrate periodic singular optical soliton. Figure 7(g)–(i) depicts the imaginary value graphs of y 3,15 ( x , t ) , revealing periodic-singular optical solitons same to the real value graphs but with variations in amplitude and phase component under specified constants: α = 0.1 , a = 0.99 , β = 1 , g 1 = − 0.5 , p 0 = 0.1 , p 2 = 0.2 , v = 0.5 , and g 0 = 1 .

Figure 8

Comparison of the original soliton (solid blue) and perturbed soliton (dashed red) showing both the slowdown effect and radiation emission. The perturbed soliton shows a backward shift due to the slowdown, and small oscillatory waves are emitted as radiation.

The inclusion of TOD is crucial for accurately modeling soliton behavior in dispersive media. Without considering the effects of TOD, the analysis is incomplete, as it neglects the physically important phenomena of soliton slowdown and radiation. The perturbative approach used here captures these effects and aligns with the expected behavior of solitons in the presence of higher-order dispersion, as shown in Figure 8.

8 Conclusion

This study has delved into an exploration of the NLSHE, which incorporates CD and serves as a governing model for comprehending the dynamics of dispersive pulse propagation in optical fibers. Employing three distinct methodologies, namely, the 1 φ ( η ) , φ ′ ( η ) φ ( η ) method, the new Kudryashov method, and the extended simple equation method, we have successfully derived solutions for this equation. Visualization of the results through 3D and 2D graphs, illustrating absolute, real, and imaginary values, provides a comprehensive understanding of soliton dynamics. These graphical representations underscore the existence of diverse soliton solutions, encompassing dark, bright, singular, periodic singular, dark–bright, and dark-singular solitons. The implications of these findings extend to the field of fiber–optic communication, where they significantly contribute to advancing our insights into the intricate dynamics of solitons. This research holds significant importance and may pave the way for further advancements and applications in the field of fiber–optic communication. By perturbing the soliton solution to include the effects of TOD, we have shown that solitons experience a gradual deceleration and emit small radiative waves. These effects are consistent with the underlying physics of dispersive systems, making the analysis more comprehensive and physically realistic.

Acknowledgments

The authors gratefully acknowledge the financial support provided by the National Science and Technology Council (NSTC), Taiwan, under Grant No. 112-2115-M-006-002.

Funding information: This work was supported by the National Science and Technology Council (NSTC), Taiwan, under Grant No. 112-2115-M-006-002.
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: All data generated or analyzed during this study are included in this published article.

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Received: 2024-09-02

Revised: 2024-10-30

Accepted: 2024-11-05

Published Online: 2025-04-29

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0058

Keywords for this article

nonlinear Schrödinger equation; optical soliton; chromatic dispersion; (1/φ(η), φ′(η)/φ(η)) method; new Kudryashov method; extended simple equation method

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