EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6764
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Novel Quasi-Periodic Type Optical Solitons and the
Formation of Fractal Structures in Non-integrable

Nonlinear Helmholtz Equations with Phase Portraits
and Chaotic Analysis

Khaled Suwais1, Nabil Mlaiki2, Shoaib Barak3, Rashid Ali4,∗

1 Faculty of Computer Studies, Arab Open University, Riyadh 11681, Saudi Arabia
2 Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi
Arabia
3 Department of Mathematics, Government Degree College Number 2, Mardan 23200, Pak-
istan
4 School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, P.R. China

Abstract. In this study, we consider new optical soliton solutions of one of the most important
non-integrable model arising in optical fibres, namely Nonlinear Helmholtz equations (NHEs) that
describes transverse interactions, transmission of coupled waves and optical solitons’ propagation in
the field of fiber optics. We apply an adapted method to obtain some novel plethora of optical quasi-
periodic soliton solutions. These solutions are presented in the shape of exponential, hyperbolic,
trigonometric and rational functions. A set of 3D visualization, contour plots and 2D curves of these
solutions physical relevance are presented with implications for the nonlinear optics. These figures
also reveal that the established optical solitons exhibit quasi-periodicity due to the combination of
linear periodic and axial perturbations, and that the presence of quasi-periodical perturbations of
the solitons leads to the formation of the fractal-like structures. We also study the chaotic/periodic
and bifurcation behavior, associated with the model, in the light of Hamiltonian analysis, as a
consequence, we find positive results of the quasi-periodicity and fractal-like structures in the
systems under consideration. Apart from offering novel analytical perspectives for dealing with
the coupled NHEs, the present results would also be a concrete contribution to the understanding
the soliton wave dynamics in complicated nonlinear media.
2020 Mathematics Subject Classifications: 35Q55, 37N20, 37G35
Key Words and Phrases: Nonlinear Helmholtz equations, unified method, complex structured
partial differential equations, optical fractals, quasi-periodic soliton, phase portraits, chaotic anal-
ysis

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6764

Email addresses: khaled.suwais@arabou.edu.sa (K. Suwais),
nmlaiki@psu.edu.sa, nmlaiki2012@gmail.com (N. Mlaiki), shoaibbarak2015@gmail.com (S. Barak),
rashidali0887@gmail.com (R. Ali)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 2 of 25

1. Introduction

Nonlinear Partial Differential Equations (NPDEs) are applied in many areas of science
[1]-[5]. Notable among them are the Higgs model in particle physics, the Drinfeld-Sokolov-
Wilson system found in fluid dynamics, the Duffing equation for periodic motions the
nonlinear Maccari’s system in hydrodynamics, the Gilson-Pickering equation in theory of
crystal lattice and plasma physics, the Kairat equations in optics, coupled NHEs in optics
etc. Several analytical and numerical techniques with exact and numerical solutions are
introduced by the academicians to study the mechanism of NPDEs. However, the NPDEs
are far from being easy, consider few obstacles spanning from single solution and the part-
ner who gets none to the sensitivity to initial conditions, the lack of a precise definition
of the exact solution and the complexity of nonlinearity. In the class of exact solutions
the case of traveling wave solutions is of special interest, in particular, solitonic solutions
provide useful information on the dynamical behavior of nonlinear models. In order to
investigate the soliton theory of such non-linear models having the ability to provide infor-
mation of underlying physical phenomena, a number of efficient methods to obtain soliton
solutions have been developed in the last few years. These methods are based on Mat-
lab, Maples, Mathematica and other symbolic computer systems that ease complicated
algebraic calculations. These include extended direct algebraic method [6], tanh-coth
approach [7], Poincar’e-Lighthill-Kuo technique [8], sine-cosine strategy [9], Jacobi ellip-
tic function methodology [10], (G’/G)-expansion approach [11]-[13], Khater technique [14],
Hirota bilinear approach [15], sub-equation strategy [16], Kudryashov technique [17], mod-
ified simple equation approach [18], first integral technique [19], unified approach [20, 21]
and so on [22]-[25].

The unified method [20, 21], which is remarkably effective and reliable, is used here to de-
rive and investigate a special class of optical soliton solutions for coupled NHEs. As some
published methods, such as the Hirota bilinear method and (G′/G)-expansion technique
and so on, have been widely utilized to construct the soliton solutions of NPDEs which lead
to complex algebraic computations. The unified method is one among some other simple
and great direct algebraic methods which produces a great deal of new soliton solutions
in the form of rational, exponential, trigonometric and hyperbolic functions. This simple
ansatz also in no need of heavy machinating e.g linearization, perturbation and many
expressed there in the literature in other approaches. The simplicity and effectiveness
of the employed unified method enable us to develop the accurate closed-form solutions
without the inconvenience of more complicated means. The model is transformed to a
system of non-linear algebraic equation in the context of series form solution by passing
the NPDEs to NODEs employing a wave transformation. Various soliton solutions can
be derived by taking any computer algebra system to solve the resultant system. A single
travelling wave packet such as a soliton is a pulse that maintains its shape and velocity in
a medium without diffusing, and one of these configurations that may maintain itself in
the time-evolution of the field is the pulse graph. From the mathematical point of view,
the soliton solutions of NPDEs are still significant, since they are much more detailed and



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 3 of 25

accurate compared with the ordinary solutions. Their natural resilience and strength also
allows them to be used useful in many technological and scientific applications. They are
very effective to preserve a large amount of concordance and for a sufficient transfer of
information as expressiveness for nonlinear systems.

In order to enhance of the phase and the modulated amplitude of continuous waves and to
simulate the gradually more compact support of plasmon resonance and photonic devices,
the NHEs makes a demand of a very important Shr”odinger-class model. From the physi-
cal point of view coupled NHEs describes transverse interactions, transmission of coupled
waves and optical solitons’ propagation in the field of fiber optics [26]. The associated
NHEs are given by [27]:

iEt +
Exx

2
+ χEtt + ς1

∣∣E2
∣∣E + ς2E

∣∣U2
∣∣ = 0,

iUt +
Uxx

2
+ χUtt + ς1

∣∣E2
∣∣U + ς2U

∣∣U2
∣∣ = 0,

(1)

where i stands for the imaginary unit whereas E = E(t, s) and U = U(t, s) are the enve-
lope fields of the first and the second component, respectively. In above system, Helmholtz
nonparaxiality is accounted for by the seconds terms Exx and Uxx along with the coeffi-
cient χ(> 0) that represents the level of the nonparaxial parameter and can be modeled as
such, while ς1 and ς2 are, nonlinearity coefficients, which can, with some stress, be placed
in the equation itself. But allowing for a more general class of nonlinearity of mixed
(focusing-defocusing) type, (or with focusing and defocusing indices, as those would ap-
pear in the focusing case also (ς1,ς2 > 0) and defocusing case (ς1,ς2 < 0), respectively.
When ς1 = ς2 = ±1, the above system reduces to Helmholtz-Manakov system.

Other investigators have also studied symmetric and asymmetric coupled NHEs before
initiating this study. For instance, Tamilselvan et al. constructed and addressed this
model through the ansatz technique [28]. Singh et al. utilized exp-function expansion
method to arrive at travelling wave solutions for the targeted model [29]. Saha et al.
developed chirped gray and anti-dark solutions for coupled NHE with cubic nonlinear-
ity [30]. Finally, by applying the (G’/G)-expansion approach, Alsaud et al. constructed
optical soliton solutions for the aimed CNHEs [27]. For the aimed NHEs, this research
exploration presents and analyzes new types of optical soliton solutions in the shape of
the exponential, hyperbolic, trigonometric and rational functions with the aid of unified
approach. The present work is motivated by the ongoing investigations of solitary wave
particularly soliton solutions of the model. To reveal as well as visualize the propagation
features of the established optical soliton solutions, this study exhibits a set of 3D, 2D
and contour plots. These illustrations demonstrate that the generated quasi-periodic type
solitons consist of periodic and axial perturbations, leading to optical fractals. Moreover,
our implemented unified method is demonstrated to be useful, giving a useful insight on
strongly coupled NHE behaviour, increasing the class of optical soliton solutions and sug-
gests applications in nonlinear model control.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 4 of 25

Finally, the study of the dynamic behavior of the systems through time series plots,
phase space diagrams as well as Hamiltonian analysis has emerged in the recent years
as a new field of study. For example, Borhan et al. investigated the bifurcation behaviour,
sensitivity analysis and chaotic motions of two (3+1)-dimensional NPDEs such as Jimbo-
Miwa equations and Kadomtsev-Petviashvili equation [31]. Hosseini et al. investigated the
dynamical system of the generalized Schrödinger equation obtained by Galilean transfor-
mation and bifurcation of the governing model using the planar dynamical system theory
[32]. In another work, via expressing equation as Hamiltonian system, Qi et al. inves-
tigated the bifurcation and phase diagrams of the electrical transmission line model in
fractional order and analyzed chaotic behavior and sensitivity of the system [33]. Finally,
Hossain et al. performed sensitivity and bifurcation analysis the Hamiltonian amplitude
equation [34]. Motivated by recent work about Hamiltonian analysis, we also investigate
the bifurcations and chaotic behavior and find that it appears in the dynamical system
and get good outcomes concerning fractals and periodicity in the governing system.

The rest of the documentation is organized in the following form: Section 2: Explains
the structure and functioning of the hired unified approach, Section 3: Establishes novel
plethora optical soliton solutions for coupled NHEs, Section 4: Graphical illustrations of
the obtained optical quasi-periodic solitons, Section 5: Presents analysis of the model with
regards to bifurcation and chaos theory and the final section presents conclusion of the
work.

2. The Working Methodology of Unified Approach

Considering the subsequent general NPDE [20, 21]:

M(E,Et, Ex1 , Ex2 , EEx1 , . . .) = 0, (2)

where E = E(t, x1, x2, x3, . . . , xk). In accordance with the unified approach, (2) is initially
transformed to the subsequent NODE with the use of wave transformation of the form
E(t, x1, x2, x3, . . . , xk) = E(ζ) where ζ:

N(E,E′E,E′, . . . ) = 0, (3)

where E′ = dE
dζ . When applicable, the integrating equation (3) can be invoked to enforce

the homogeneous balance condition on the NODE. Next, the close-form solution for the
resulting NODE in (3) is then supposed as:

E(ζ) =

P∑
j=−P

αj(V (ζ))j , (4)

where V (ζ) is determined by the following Riccati equation, αj(j = −P, ..., P ) are con-
stants that should be determined by calculation, and P ∈ N is known as balance number



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 5 of 25

that is derived from the homogeneous balancing of the largest nonlinearity and the highest-
order derivative in (3):

V ′(ζ) = (V (ζ))2 + β. (5)

Where V ′ = dV
dζ and β is a real constant. Moreover, corresponding to the values of β, we

have the ensuing classes of exact solutions for (5):
Type. 1: Considering β < 0:

V1(ζ) = ±
√

− (ϑ2 + κ2)β − ϑ
√
−β cosh

(
2
√
−β (ζ + ξ)

)
ϑ sinh

(
2
√
−β (ζ + ξ)

)
+ κ

, (6)

V2(ζ) = ±
√
−β +

∓2ϑ
√
−β

ϑ+ cosh
(
2
√
−β (ζ + ξ)

)
∓ sinh

(
2
√
−β (ζ + ξ)

) , (7)

V3(ζ) =
√
−β tanh

(√
−β (ζ + ξ)

)
, (8)

and
V4(ζ) =

√
−β coth

(√
−β (ζ + ξ)

)
. (9)

Type. 2: Considering β > 0:

V5(ζ) =
±
√
(ϑ2 − κ2)β − ϑ

√
β cos

(
2
√
β (ζ + ξ)

)
ϑ sin

(
2
√
β (ζ + ξ)

)
+ κ

, (10)

V6(ζ) = ±i
√
β +

∓2 iϑ
√
β

ϑ+ cos
(
2
√
β (ζ + ξ)

)
∓ sin

(
2
√
β (ζ + ξ)

) , (11)

V7(ζ) =
√
β tan

(√
β (ζ + ξ)

)
, (12)

and
V8(ζ) = −

√
β cot

(√
β (ζ + ξ)

)
. (13)

Type. 3: Considering β = 0:
V9(ζ) =

−1

ζ + ξ
. (14)

In above solutions ξ, ϑ, κ ∈ R. A system of algebraic equations is then produced
when (4) is incorporated into (3) and the coefficients of V (ζ) are equated to zero. When
solved using Maple, this system yields values of αj(j = −P, ..., P ) and other unknown.
When substituted these acquired values (along with the solution V (ζ) of (5))in (4), explicit
solutions for (2) are established.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 6 of 25

3. Main Results

The proposed unified approach is used in this section of the investigation to establish
novel plethora of optical soliton solutions for coupled NHEs in (1). Initially, the ensuing
complex transformation is applied to (1):

E(t, s) = eiΥE(ζ), U(t, s) = eiΥU(ζ),

ζ = (µ3s− µ3µ4t), Υ = (−µ1s+ µ2t+ θ).
(15)

This transformation yields

(−µ3
2 − 2χµ3

2µ4
2)E′′ + (µ1

2 + 2µ2 + 2χµ2
2)E+ (−2ς1E

3 − 2ς2EU
2) = 0,

(2µ1µ3 + 2µ3µ4 + 4χµ2µ3µ4)E
′ = 0,

(16)

from the real component whilst the imaginary part provides:

(−µ3
2 − 2χµ3

2µ4
2)U′′ + (µ1

2 + 2µ2 + 2χµ2
2)U+ (−2ς1E

2U− 2ς2U
3) = 0,

(2µ1µ3 + 2µ3µ4 + 4χµ2µ3µ4)U
′ = 0.

(17)

The entire system of NODEs given in (16) and (17) is reduced to the ensuing single NODE
when U = E is considered:

Θ1E+Θ2E
′′ +Θ3E

3 = 0, (18)

Also from the imaginary part, we have the following constraint condition:

µ4 =
−µ1

1 + 2χµ2
. (19)

Also in (18)

Θ1 = (µ1
2 + 2µ2 + 2χµ2

2),

Θ2 = (−µ3
2 − 2χµ3

2µ4
2),

Θ3 = (−2(ς1 + ς2)).

By homogeneously balancing E3(ζ) with E′′(ζ) presented in (18), we get P = 1 which
suggests the subsequent solution for (18) when substituted in (4):

E(ζ) =
1∑

j=−1

αj(V (ζ))j . (20)

Substituting (20) for ζ into (18), combining every term in V (ζ) in the equal power we
obtain expression in V (ζ). The problem is reduced to the following system of nonlinear
algebraic equations when the coefficients are set to zero:

− 2µ3
2α1 − 2 ς2α1

3 − 2 ς1α1
3 − 4

χµ3
2µ1

2α1

(1 + 2χµ2)
2 = 0,



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 7 of 25

− 6 ς2α0α1
2 − 6 ς1α0α1

2 = 0,

(−6 ς1α−1 − 6 ς2α−1)α1
2 +

(
−4χµ3

2µ1
2β

(1 + 2χµ2)
2 − 6 ς2α0

2 − 2µ3
2β − 6 ς1α0

2 + 2χµ2
2 + µ1

2 + 2µ2

)
α1 = 0,

2µ2α0 − 12 ς2α−1α0α1 + µ1
2α0 − 12 ς1α−1α0α1 − 2 ς2α0

3 + 2χµ2
2α0 − 2 ς1α0

3 = 0,

(−6 ς2α1 − 6 ς1α1)α−1
2 +

(
−4χµ3

2µ1
2β

(1 + 2χµ2)
2 − 6 ς2α0

2 − 2µ3
2β − 6 ς1α0

2 + 2χµ2
2 + µ1

2 + 2µ2

)
α−1 = 0,

− 6 ς2α−1
2α0 − 6 ς1α−1

2α0 = 0,

and

− 4
χµ3

2µ1
2α−1β

2

(1 + 2χµ2)
2 − 2 ς1α−1

3 − 2µ3
2α−1β

2 − 2 ς2α−1
3 = 0.

Three (3) families of solutions are obtained by Maple for the resulting problem:
Case. 1

α0 = 0, α1 = 0, α−1 = α−1, µ1 =

√
− −2χµ2

2 − 2µ2 + 2µ3
2β

4µ3
2β χ− 1− 4χµ2 − 4χ2µ2

2
(1 + 2χµ2) , µ2 = µ2,

µ3 = µ3, χ = χ, ς1 =
4 ς2α−1

2χ2µ2
2 + 4 ς2α−1

2χµ2 − 4µ3
2β χ ς2α−1

2 + ς2α−1
2 + µ3

2β2

α−1
2 (4µ3

2β χ− 1− 4χµ2 − 4χ2µ2
2)

, ς2 = ς2.

(21)

Case. 2

α0 = 0, α1 = α1, α−1 = 0, µ1 =

√
− −2χµ2

2 − 2µ2 + 2µ3
2β

4µ3
2β χ− 1− 4χµ2 − 4χ2µ2

2
(1 + 2χµ2) , µ2 = µ2,

µ3 = µ3, χ = χ, ς1 = −−4 ς2α1
2χ2µ2

2 − 4 ς2α1
2χµ2 + 4µ3

2β χ ς2α1
2 − ς2α1

2 − µ3
2

α1
2 (4µ3

2β χ− 1− 4χµ2 − 4χ2µ2
2)

, ς2 = ς2.

(22)

Case. 3

α0 = 0, α1 = α1, α−1 = β α1, µ1 =

√
− 2χµ2

2 + 2µ2 + 4µ3
2β

8µ3
2β χ+ 1 + 4χµ2 + 4χ2µ2

2
(1 + 2χµ2) , µ2 = µ2,

µ3 = µ3, χ = χ, ς1 = −4 ς2α1
2χ2µ2

2 + 4 ς2α1
2χµ2 + 8µ3

2β χ ς2α1
2 + ς2α1

2 + µ3
2

α1
2 (8µ3

2β χ+ 1 + 4χµ2 + 4χ2µ2
2)

, ς2 = ς2.

(23)

Using Case. 1 together with the solution to (4), (15) and (20), we establish the
subsequent novel class of solutions for coupled NHEs stated in (1):
Type. 1.1 Considering β < 0:

E1,1(t, s) = U1,1(t, s) =eiΥ
(

α−1

(
ϑ sinh

(
2
√
−β (ζ + ξ)

)
+ κ
)√

− (ϑ2 + κ2)β − ϑ
√
−β cosh

(
2
√
−β (ζ + ξ)

)), (24)



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 8 of 25

E1,2(t, s) = U1,2(t, s) =eiΥ
(
α−1

(√
−β − 2

ϑ
√
−β

ϑ+ cosh
(
2
√
−β (ζ + ξ)

)
− sinh

(
2
√
−β (ζ + ξ)

))−1)
,

(25)

E1,3(t, s) = U1,3(t, s) =eiΥ
(

α−1√
−β tanh

(√
−β (ζ + ξ)

)), (26)

and

E1,4(t, s) = U1,4(t, s) =eiΥ
(

α−1√
−β coth

(√
−β (ζ + ξ)

)). (27)

Type. 1.2 Considering β > 0:

E1,5(t, s) = U1,5(t, s) =eiΥ
(

α−1

(
ϑ sin

(
2
√
β (ζ + ξ)

)
+ κ
)√

(ϑ2 − κ2)β − ϑ
√
β cos

(
2
√
β (ζ + ξ)

)), (28)

E1,6(t, s) = U1,6(t, s) =eiΥ
(
α−1

(
i
√
β − 2 iϑ

√
β

ϑ+ cos
(
2
√
β (ζ + ξ)

)
− sin

(
2
√
β (ζ + ξ)

))−1)
,

(29)

E1,7(t, s) = U1,7(t, s) =eiΥ
(

α−1√
β tan

(√
β (ζ + ξ)

)), (30)

and

E1,8(t, s) = U1,8(t, s) =eiΥ
(
− α−1√

β cot
(√

β (ζ + ξ)
)). (31)

Type. 1.3 Considering β = 0:

E1,9(t, s) = U1,9(t, s) =eiΥ
(
− α−1 (ζ + ξ)

)
. (32)

Where ζ = µ3(s− (
−
√

− −2χµ2
2−2µ2+2µ3

2β

4µ3
2β χ−1−4χµ2−4χ2µ2

2 (1+2χµ2)

1+2χµ2
)t),

Υ = (−
√
− −2χµ2

2−2µ2+2µ3
2β

4µ3
2β χ−1−4χµ2−4χ2µ2

2 (1 + 2χµ2)s+ µ2t+ θ).

Using Case. 2 together with the solution to (4), (15) and (20), we establish the subsequent
novel class of solutions for coupled NHEs stated in (1):
Type. 2.1 Considering β < 0:

E2,1(t, s) = U2,1(t, s) =eiΥ
(α1

(√
− (ϑ2 + κ2)β − ϑ

√
−β cosh

(
2
√
−β (ζ + ξ)

))
ϑ sinh

(
2
√
−β (ζ + ξ)

)
+ κ

)
, (33)



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 9 of 25

E2,2(t, s) = U2,2(t, s) =eiΥ
(
α1

(√
−β − 2

ϑ
√
−β

ϑ+ cosh
(
2
√
−β (ζ + ξ)

)
− sinh

(
2
√
−β (ζ + ξ)

))),
(34)

E2,3(t, s) = U2,3(t, s) =eiΥ
(
α1

√
−β tanh

(√
−β (ζ + ξ)

))
, (35)

and

E2,4(t, s) = U2,4(t, s) =eiΥ
(
α1

√
−β coth

(√
−β (ζ + ξ)

))
. (36)

Type. 2.2 Considering β > 0:

E2,5(t, s) = U2,5(t, s) =eiΥ
(α1

(√
(ϑ2 − κ2)β − ϑ

√
β cos

(
2
√
β (ζ + ξ)

))
ϑ sin

(
2
√
β (ζ + ξ)

)
+ κ

)
, (37)

E2,6(t, s) = U2,6(t, s) =eiΥ
(
α1

(
i
√
β − 2 iϑ

√
β

ϑ+ cos
(
2
√
β (ζ + ξ)

)
− sin

(
2
√
β (ζ + ξ)

))),
(38)

E2,7(t, s) = U2,7(t, s) =eiΥ
(
α1

√
β tan

(√
β (ζ + ξ)

))
, (39)

and

E2,8(t, s) = U2,8(t, s) =eiΥ
(
− α1

√
β cot

(√
β (ζ + ξ)

))
. (40)

Type. 2.3 Considering β = 0:

E2,9(t, s) = U2,9(t, s) =eiΥ
(
− α1

ζ + ξ

)
. (41)

Where ζ = µ3(s− (
−
√

− −2χµ2
2−2µ2+2µ3

2β

4µ3
2β χ−1−4χµ2−4χ2µ2

2 (1+2χµ2)

1+2χµ2
)t),

Υ = (−
√
− −2χµ2

2−2µ2+2µ3
2β

4µ3
2β χ−1−4χµ2−4χ2µ2

2 (1 + 2χµ2)s+ µ2t+ θ).

Using Case. 2 together with the solution to (4), (15) and (20), we establish the subsequent
novel class of solutions for coupled NHEs stated in (1):
Type. 3.1 Considering β < 0:

E3,1(t, s) = U3,1(t, s) =eiΥ
(

β α1

(
ϑ sinh

(
2
√
−β (ζ + ξ)

)
+ κ
)√

− (ϑ2 + κ2)β − ϑ
√
−β cosh

(
2
√
−β (ζ + ξ)

)
+

α1

(√
− (ϑ2 + κ2)β − ϑ

√
−β cosh

(
2
√
−β (ζ + ξ)

))
ϑ sinh

(
2
√
−β (ζ + ξ)

)
+ κ

)
,

(42)



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 10 of 25

E3,2(t, s) = U3,2(t, s) =eiΥ
(
β α1

(√
−β − 2

ϑ
√
−β

ϑ+ cosh
(
2
√
−β (ζ + ξ)

)
− sinh

(
2
√
−β (ζ + ξ)

))−1

+ α1

(√
−β − 2

ϑ
√
−β

ϑ+ cosh
(
2
√
−β (ζ + ξ)

)
− sinh

(
2
√
−β (ζ + ξ)

))),
(43)

E3,3(t, s) = U3,3(t, s) =eiΥ
(

β α1√
−β tanh

(√
−β (ζ + ξ)

) + α1

√
−β tanh

(√
−β (ζ + ξ)

))
,

(44)

and

E3,4(t, s) = U3,4(t, s) =eiΥ
(

β α1√
−β coth

(√
−β (ζ + ξ)

) + α1

√
−β coth

(√
−β (ζ + ξ)

))
.

(45)

Type. 3.2 Considering β > 0:

E3,5(t, s) = U3,5(t, s) =eiΥ
(

β α1

(
ϑ sin

(
2
√
β (ζ + ξ)

)
+ κ
)√

(ϑ2 − κ2)β − ϑ
√
β cos

(
2
√
β (ζ + ξ)

)
+

α1

(√
(ϑ2 − κ2)β − ϑ

√
β cos

(
2
√
β (ζ + ξ)

))
ϑ sin

(
2
√
β (ζ + ξ)

)
+ κ

)
,

(46)

E3,6(t, s) = U3,6(t, s) =eiΥ
(
β α1

(
i
√
β − 2 iϑ

√
β

ϑ+ cos
(
2
√
β (ζ + ξ)

)
− sin

(
2
√
β (ζ + ξ)

))−1

+ α1

(
i
√

β − 2 iϑ
√
β

ϑ+ cos
(
2
√
β (ζ + ξ)

)
− sin

(
2
√
β (ζ + ξ)

))),
(47)

E3,7(t, s) = U3,7(t, s) =eiΥ
(

α1
√
β

tan
(√

β (ζ + ξ)
) + α1

√
β tan

(√
β (ζ + ξ)

))
, (48)

and

E3,8(t, s) = U3,8(t, s) =eiΥ
(
− α1

√
β

cot
(√

β (ζ + ξ)
) − α1

√
β cot

(√
β (ζ + ξ)

))
. (49)

Type. 3.3 Considering β = 0:

E3,9(t, s) = U3,9(t, s) =eiΥ
(
− α1

ζ + ξ

)
. (50)



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 11 of 25

Where ζ = µ3(s− (
−
√

− 2χµ2
2+2µ2+4µ3

2β

8µ3
2β χ+1+4χµ2+4χ2µ2

2 (1+2χµ2)

1+2χµ2
)t),

Υ = (−
√
− 2χµ2

2+2µ2+4µ3
2β

8µ3
2β χ+1+4χµ2+4χ2µ2

2 (1 + 2χµ2)s+ µ2t+ θ).

4. Graphs and Discussion

In this parts of this study, the established optical soliton solutions are shown and dis-
cussed graphically that were observed in the model. We have extracted and visualized
wave dynamics of optical solitons in 2D, 3D and contour forms with the help of unified
technique. These concepts are essential to understand the behavior of connected physical
processes. Optical pulse theory is expected to be greatly enriched from optical soliton
solutions generated. It was also demonstrated that the optical quasi-periodic solitons
corresponding to the coupled NHEs have been formally, visualized through the chaotic
perturbation that are axial-periodic perturbations.

The representations also exhibited the existence of the axially-periodic perturbing phe-
nomena in the generated solitons which led to the formation of fractals. These waveforms
have the celebrated property that after scattering or interacting with other identical wave-
forms they have a tendency to spontaneously repair themselves and settle down to be
stable. The resultant soliton structures in the NHEs are concrete with respect to the
advanced optical technologies. In optical fiber communication systems, solitons exhibit
themselves as the forces to neutralize nonlinearity and dispersion, being a core constituent
of long-distance and high-capacity signal transport. Fractal soliton profiles, induced by
periodic and axial perturbations, demonstrate their intimate relation to localization of
light in the fiber optic and to fractal generation of optical fields, which can be useful for
designing optical sensors with sensitivity limit extended down to the level of single atoms
and for fabrication of optical lattices.

The found quasi-periodic solitons can also be used for all optical reshaping and buffer-
ing of signals which is important for all optical computing.

New trends in soliton-theory has created an interesting possibility for new mathe-
matical topics, such as the so called fractal soliton theory [35]-[37] which is the complex
relationships between solitons and fractals. A fractal solitary wave is a stable localized
wave packet with a simultaneous fractal and a solitonic structure. These results can be con-
nected to solitons and geometric fractals to understand nonlinear phenomena in physics,
engineering, biology [38]-[40]. Fractal solitons are important tools when addressing chaos
theory since a special feature allows us to understand the dynamics and the stability of
chaotic systems. Relating the attention attracted by research on fractal soliton in recent
mathematical enterprises of physical interests will generate some new thinking with re-
spect to theoretical and potential mathematics.

Figure. 1 reveals the formation of fractal structure due to the propagation of quasi-



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 12 of 25

periodic type optical soliton represented by E1,1(t, s) described in (24) for µ2 = 0.005, µ3 =
0.002, β = −1, α−1 = 2, ϑ = 25, κ = 3, ξ = 1, χ = 1, θ = 0.5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 0.
Figure. 2 reveals the formation of fractal structure due to the propagation of quasi-periodic
type optical soliton represented by E1,3(t, s) described in (26) for µ2 = 0.0015, µ3 =
0.0025, β = −4, α−1 = 5, ϑ = 4, κ = 1, ξ = 2, χ = 3, θ = 1. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 10.
Figure. 3 reveals the formation of fractal structure due to the propagation of quasi-periodic
type optical soliton represented by E1,5(t, s) described in (28) for µ2 = 0.0075, µ3 =
0.0015, β = 5, α−1 = 2, ϑ = 3, κ = 2, ξ = 1, χ = 2, θ = 5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 50.
Figure. 4 reveals the formation of fractal structure due to the propagation of quasi-periodic
type optical soliton represented by E2,2(t, s) described in (34) for µ2 = 0.0035, µ3 =
0.0055, β = −3, α1 = 1, ϑ = 4, κ = 2, ξ = 1, χ = 3, θ = 5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 50.
Figure. 5 reveals the formation of fractal structure due to the propagation of quasi-periodic
type optical soliton represented by E2,6(t, s) described in (38) for µ2 = 0.0025, µ3 =
0.0095, β = 9, α1 = 10, ϑ = 10, κ = 1, ξ = 10, χ = 5, θ = 10. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 100.
Figure. 6 reveals the formation of fractal structure due to the propagation of quasi-periodic
type optical soliton represented by E2,9(t, s) described in (41) for µ2 = 0.00885, µ3 =
0.00775, β = 0, α1 = 5, ϑ = 9, κ = 5, ξ = 20, χ = 10, θ = 50. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 10.
Figure. 7 reveals the formation of fractal structure due to the propagation of quasi-periodic
type optical soliton represented by E3,1(t, s) described in (42) for µ2 = 0.0005, µ3 =
0.001, β = −25, α1 = 2, ϑ = 8, κ = 2, ξ = 1, χ = 1, θ = 10. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 50.
Figure. 8 reveals the formation of fractal structure due to the propagation of quasi-
periodic type optical soliton represented by E3,4(t, s) described in (45) for µ2 = 0.015, µ3 =
0.031, β = −1, α1 = 1, ϑ = 5, κ = 1, ξ = 2, χ = 0.5, θ = 5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 0.
Figure. 9 reveals the formation of fractal structure due to the propagation of quasi-
periodic type optical soliton represented by E3,8(t, s) described in (49) for µ2 = 0.005, µ3 =
0.001, β = 4, α1 = 2, ϑ = 6, κ = 5, ξ = 1, χ = 1, θ = 2. Moreover, the 2D plots of the real
and imaginary parts of the solution are displayed for s = 10.
Figure. 10 reveals the formation of fractal structure due to the propagation of quasi-
periodic type optical soliton represented by E3,9(t, s) described in (50) for µ2 = 0.125, µ3 =
0.525, β = 0, α1 = 5, ϑ = 1, κ = 0, ξ = 2, χ = 5, θ = 2. Moreover, the 2D plots of the real
and imaginary parts of the solution are displayed for s = 10.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 13 of 25

Figure 1: Propagation of quasi-periodic type optical soliton represented by E1,1(t, s) described in (24) for
µ2 = 0.005, µ3 = 0.002, β = −1, α−1 = 2, ϑ = 25, κ = 3, ξ = 1, χ = 1, θ = 0.5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 0.

Figure 2: Propagation of quasi-periodic type optical soliton represented by E1,3(t, s) described in (26) for
µ2 = 0.0015, µ3 = 0.0025, β = −4, α−1 = 5, ϑ = 4, κ = 1, ξ = 2, χ = 3, θ = 1. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 10.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 14 of 25

Figure 3: Propagation of quasi-periodic type optical soliton represented by E1,5(t, s) described in (28) for
µ2 = 0.0075, µ3 = 0.0015, β = 5, α−1 = 2, ϑ = 3, κ = 2, ξ = 1, χ = 2, θ = 5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 50.

Figure 4: Propagation of quasi-periodic type optical soliton represented by E2,2(t, s) described in (34) for
µ2 = 0.0035, µ3 = 0.0055, β = −3, α1 = 1, ϑ = 4, κ = 2, ξ = 1, χ = 3, θ = 5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 50.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 15 of 25

Figure 5: Propagation of quasi-periodic type optical soliton represented by E2,6(t, s) described in (38) for
µ2 = 0.0025, µ3 = 0.0095, β = 9, α1 = 10, ϑ = 10, κ = 1, ξ = 10, χ = 5, θ = 10. Moreover, the 2D plots of
the real and imaginary parts of the solution are displayed for s = 100.

Figure 6: Propagation of quasi-periodic type optical soliton represented by E2,9(t, s) described in (41) for
µ2 = 0.00885, µ3 = 0.00775, β = 0, α1 = 5, ϑ = 9, κ = 5, ξ = 20, χ = 10, θ = 50. Moreover, the 2D plots of
the real and imaginary parts of the solution are displayed for s = 10.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 16 of 25

Figure 7: Propagation of quasi-periodic type optical soliton represented by E3,1(t, s) described in (42) for
µ2 = 0.0005, µ3 = 0.001, β = −25, α1 = 2, ϑ = 8, κ = 2, ξ = 1, χ = 1, θ = 10. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 50.

Figure 8: Propagation of quasi-periodic type optical soliton represented by E3,4(t, s) described in (45) for
µ2 = 0.015, µ3 = 0.031, β = −1, α1 = 1, ϑ = 5, κ = 1, ξ = 2, χ = 0.5, θ = 5. Moreover, the 2D plots of the
real and imaginary parts of the solution are displayed for s = 0.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 17 of 25

Figure 9: Propagation of quasi-periodic type optical soliton represented by E3,8(t, s) described in (49) for
µ2 = 0.005, µ3 = 0.001, β = 4, α1 = 2, ϑ = 6, κ = 5, ξ = 1, χ = 1, θ = 2. Moreover, the 2D plots of the real
and imaginary parts of the solution are displayed for s = 10.

Figure 10: Propagation of quasi-periodic type optical soliton represented by E3,9(t, s) described in (50) for
µ2 = 0.125, µ3 = 0.525, β = 0, α1 = 5, ϑ = 1, κ = 0, ξ = 2, χ = 5, θ = 2. Moreover, the 2D plots of the real
and imaginary parts of the solution are displayed for s = 10.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 18 of 25

5. Phase Portraits and Time-Series Evaluation

This section provides and evaluates phase portraits of the planner system and the time
series plots of the perturbed system using the notion of the Hamiltonian analysis.

5.1. Phase Portraits and Bifurcation Analysis

Using the the notion of the bifurcation theory, we get the subsequent planner system
of (9)

E(ζ) = G(ζ)

z(H(ζ), G(ζ)) = G′(ζ) = H(ζ),

r(H(ζ), G(ζ)) = H(ζ)′ = −Θ1

Θ2
G(ζ)− Θ2

Θ2
G(ζ)3,

(51)

where

Θ1 = (µ1
2 + 2µ2 + 2χµ2

2),

Θ2 = (−µ3
2 − 2χµ3

2µ4
2),

Θ3 = (−2(ς1 + ς2)).

Under a given integral, this system displays the following Hamiltonian:

Ham(H(ζ), G(ζ)) =
H(ζ)2

2
+

Θ1

2Θ2
G(ζ)2 +

Θ2

4Θ2
G(ζ)4. (52)

We get three equilibria: (0, 0), (R1, 0) and (R2, 0) for the planner system where R1 and
R2 are given below:

R1 =

√
−Θ1

Θ3
, R2 = −

√
−Θ1

Θ3
. (53)

Furthermore, according to the Jacobian matrix:

J =

 ∂z
∂G(ζ)

∂z
∂H(ζ)

∂r
∂G(ζ)

∂r
∂H(ζ)

 , (54)

The Jacobian of the system is:

|J(H(ζ), G(ζ))| = 3Θ3

Θ2
G(ζ)2 +

Θ1

Θ2
. (55)

An equilibrium point is a saddle or a center based on the value of |J(H(ζ), G(ζ))| i.e., an
equilibrium point is a saddle for |J(H(ζ), G(ζ))|0, and transportation if |J(H(ζ), G(ζ))| =
0. We show next some phase portraits of the planner dynamical system as function of the
parameters by selecting different values for them:



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 19 of 25

[H]
Figure 11: The visualization of phase portrait for (51) at µ1 = 0.10, µ2 = 0.20, µ3 = 0.20, χ = 0.10, ς1 = 0.30, ς2 = 0.30.

Remarks: Considering Θ1/Θ2 < 0, according to the phase portrait of planner dynamical
system in Fig. (11), we obtain the homoclinic loop with centers (R1, 0), (R2, 0) and saddle
point (0, 0). Thus, the system seems to shuttle periodically in the state variable coordinate
system [G(ζ), H(ζ)]. This can be interpreted to be a regular, dynamic process, caused
by mutual emergence of different phases of soliton and exchange of energy. Likewise, the
phase portrait of planner dynamical system in the case of Θ1/Θ2 > 0 has been illustrated
in Fig. 12, showing a closed circle centered at (0, 0), implying that the system supports
a periodic soliton. Closed paths can appear in phase portrait when there is confined or
restricted periodic motion. Which indicates the presence of solitonic cycles, where the
energy of wave is confined in a repeating cycle without being dissipated rather than fading
away or spreading indefinitely.

5.2. The Perturbed System and Time-Series Plots Evaluation

This section presents the time series plots evaluation of the perturbed system for (9).
The planner system in (51) is perturbed by adding the periodic term to it for disturbing
the periodic behaviour of the system using the Gillion transformation. The perturbed
dynamical system for (9) with a periodic external forcing can be expressed as:

z(H(ζ), G(ζ)) = G(ζ)′ = H(ζ),

y(H(ζ), G(ζ)) = H(ζ)′ = −Θ1

Θ2
G(ζ)− Θ2

Θ2
G(ζ)3 + k0 sin(γζ).

(56)



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 20 of 25

[H]
Figure 12: The visualization of phase portrait for (51) at µ1 = 5, µ2 = 2, µ3 = 3, χ = −5, ς1 = 5, ς2 = 1.

Here in (56), k0 and γ are the amplitude and the frequency the applied external force,
respectively. We show the presence of periodic/chaotic behavior in the perturbed system
in (56) for (G(0) = 0,H(0) = 1) as initial condition and assigning arbitrary values to the
involved free parameter in Figures 11 and 12.
Remarks In the case of Θ1/Θ2 < 0, the time-series plot in Figure. 13 shows quasi-periodic
and fractal-like oscillations. The curve’s this periodicity may be related to such structures
as fractal-like periodic solitons will oscillate in both space and time. Analogously, the
fractal-like periodic solitons are proved by the periodic oscillations with fractal structures
at Θ1/Θ2 > 0 in Figure 14. Regular oscillations in chaotic systems prove that the system
does not disappear into total chaos, but that despite their complexity and nonlinearity,
produce predictable and regularly repeated patterns. Overall, all the time-series plots
reveals that the system in (56) is fractal-like periodic.



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 21 of 25

[H]
Figure 13: The time series plots of (56) for µ1 = 0.10, µ2 = 0.20, µ3 = 0.20, χ = 0.10, ς1 = 0.30, ς2 = 0.30, k0 =
0.2, γ = 2π.

[H]
Figure 14: The time series plots of (56) for µ1 = 5, µ2 = 2, µ3 = 3, χ = −5, ς1 = 5, ς2 = 1, k0 = 5, γ = 5π.

6. Conclusion

In this work we have constructed some exact optical quasi-periodic soliton solutions
for coupled NHEs via the effective unified method. We established rational, exponential,
trigonometric and hyperbolic solutions. To show and explain the dynamical evolutions of
the constructed optical solitons, we provided several 3D, 2D, and contour graphs under
choices of the involved free parameters. These figures showed that the periodic and axial
perturbations in the obtained quasi-periodic type solitons degenerate optical fractals. We
also studied the Hamiltonian nature that gave positive fallout for quasi-periodicity and
fractals for the governing system. The proposed optical solitons are expected to have
many useful applications in the communication field. It is also worth noting that our used
unified technique demonstrates its worth via presenting significant information of the cou-
pled NHE dynamics, enlarging the family of optical soliton structures, and also unveiling
potential applications in the field of nonlinear model management. It is important to



K. Suwais et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6764 22 of 25

remember that the unified approach, however enlightening, has some limitations, even if
it provided a significantly improved understanding of the soliton dynamics and of its rela-
tion to the models. Especially, the method fails if the highest order of the derivative and
the nonlinear part are not uniformly balanced. Nevertheless, from the results, it can be
concluded that the applied unified method is the one capable for various nonlinear model
and it also relaxing as well as convenient for this kind of model. A future objective of the
current work is to investigate soliton solutions for the proposed model in a stochastic and
fractional context.

Funding

The authors extend their appreciation to the Arab Open University for funding this
work through AOU research fund no. (AOUKSA-524008).

Acknowledgements

The author Khaled Suwais is grateful to the Arab Open University for facilitating this
research work. The author N. Mlaiki would like to acknowledge Prince Sultan University
for providing support through the TAS research lab.

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