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This study explores the Triki-Biswas (TB) model, a novel model describing soliton dynamics in monomodal optical fibers with non-Kerr dispersion, to obtain optical solitons. Optical bright and singular solitons were derived using the generalized Jacobi elliptic function (gJEF) method and the expansion method. Trigonometric, hyperbolic, exponential, polynomial, and rational functions are obtained. The physical dynamics of the obtained solutions confirmed the existence of known complex structures, such as shock waves, dark solitons, periodic waves, and singular periodic solutions. The simulations generated in Mathematica 11.3 are graphically presented to depict the nature of the acquired solutions. These results are novel and have not been reported previously in the literature.
Solitons1,2 play a pivotal role in soliton transmission technology3–11, particularly in applications involving optical fibers12–15, telecommunications, and data transmission16–19 across transcontinental and transoceanic distances. Numerous mathematical models20–25, including but not limited to the complex Ginzburg-Landau model, Fokas-Lenells equation, Radhakrishnan-Kundu-Lakshmanan equation, Lakshmanan-Porsezian-Daniel model, Kundu-Eckhaus model, Kaup-Newell equation, nonlinear Schrödinger’s equation, and Gerdjikov-Ivanov equation, contribute to the comprehension and manipulation of solitons in these optical contexts26–37 and many others38–42. The TB equation is another crucial governing model employed in various techniques, such as chirped soliton solutions, the exp-expansion technique, conservation laws, first integral technique, and traveling wave hypothesis43–45. Numerous46–48 other studies exist in the literature irrespective of conservation laws.
The TB equation represents a significant advancement and serves as a generalized form of the derivative nonlinear Schrödinger equation. This equation is specifically tailored to govern the dynamics of subpicosecond pulse propagation. Notably, the TB model is a promising candidate for describing the propagation of ultrashort pulses in optical fiber systems, particularly in scenarios where the Kerr effect imposes limitations. The incorporation of derivative quintic non-Kerr nonlinearity terms within this model plays a pivotal role, especially in facilitating the transmission of extremely brief pulses with widths of the order of sub-10 fs in highly nonlinear optical fibers. Given the challenges faced by the telecommunications industry, the TB equation has emerged as a valuable asset that significantly contributes to the generation of essential optical solitons. Numerous studies have been conducted on the TB model49–51.
The TB model is investigated by employing the generalized Jacobi elliptic function method52–54 and -expansion method55,56. The primary objective is to recover subpicosecond optical soliton solutions and ascertain the conditions that govern their existence. Additionally, the adopted methods led to the discovery of supplementary solutions, including shock waves, double periodic waves, and singular periodic solutions, facilitated by the reverse formulation of the constraints. A comprehensive analysis of the model’s intricacies is presented in subsequent sections of this article. None of the ansatz methods are so strong that they can deal with all types of solutions for each NLPDE. The generalized Jacobi elliptic function method does not apply to nonlinear problems/PDEs, where the product of the even and odd terms appears as a single term. This section covers the remaining cases. In addition, it is very difficult to deal with some classes of variable coefficient NLPDEs using both techniques.
The remainder of this paper is organized as follows. In "Coordinated strategies" section, comprehensive methodologies for the gJEF method and -expansion method are presented. The application of these techniques to the TB equation is described in "Solitary wave solutions in the TB model (1)" section. In addition to the mathematical derivations, "Analysis of the physical implications of the obtainedresults" section provides a graphical representation of the outcomes, aiding the interpretation of their physical significance.Thee paper concludes with a discussion and concluding remarks in "Discussion and conclusions" section.
Formulation of the regulatory model
The model proposed by Triki and Biswas43–45 is presented as follows
1
The initial term in the equation governs the temporal evolution of pulses with the coefficient ‘, ensuring the presence of group velocity dispersion in the model. The profile of subpicosecond optical solitons is represented by the complex-valued function Q(x, t). The non-Kerr dispersion effect is counteracted by coefficient ‘’ when . When the nonlinearity parameter takes the value of , the model aligns with the Kaup-Newell model. Conversely, when , the significance of the derivative quintic non-Kerr nonlinearity terms becomes pronounced in the transmission of extremely short pulses, characterized by widths around sub-10 fs, within highly nonlinear optical fibers.
Coordinated strategies
Examine the nonlinear PDE expressed in the following form
2
where denotes the solution of the nonlinear PDE (2). Using this transformation, we obtain
3
where the parameters represent the soliton frequency, denotes the soliton wave, signifies the soliton phase, and c represents the speed of the wave. Then the nonlinear PDE (2) can be transformed into an ordinary differential equation (ODE) as follows
4
where
General procedure to the gJEF method.
In this scenario, the gJEF method was detailed using the following approach: To arrive at waveform solutions for Eq (2), it is essential to follow these specified steps;
Step 1: Take into account the subsequent structure as the solution for Eq (4);
5
where, the identification of the real parameters is necessary and the function satisfies the solution
6
where and are parameters.
Step 2: The parameter N can be determined by using the homogeneous balancing principle.
Step 3: Upon substituting Eq (5) into Eq (4) and then using Eq (6), we derive an associated system of equations featuring various monomials. Solving this system yields a set of values for the required parameters.
Step 4: The constants , , and values presented in Table 1 can be employed to deduce solutions for Eq (6).
Balance index N can be determined using the homogeneous balance principle.
Step: 3 Upon obtaining the value of N in the previous step, substitute Eq (4), and the coefficients of and . A system of algebraic equations was derived by setting each coefficient to zero. When solved using Mathematica software, these equations allow for the determination of the values of , , , , , and .
Step: 4 Substitute the values of , , , ..., , , and c into Eq (8), the solution for ODE (4) is obtained. The solution for PDE (2) follows by using the transformation (3).
In order to obtain exact solution to Eq (1), we apply the traveling wave transformation (3), and subsequently separating real and imaginary parts results
28
29
Both the real and imaginary components describe the speed of the model through the medium by the relation , so one can get
30
The homogeneous balancing principle suggests the index for Eq (30)
31
Soliton solutions using the gJEF method
In this section, the solitary wave and periodic solutions for the TB model (1) are calculated. We employ the gJEF method to handle these waveform solutions. For , the Eq (5) suggests
32
We insert the values in (30) and subsequently use (6) to arrive at the system of equations. We solve this system using Mathematica and follow the results
33
By substituting the values of parameters, the solution (32) becomes as
34
For different values of function , (34) ascertains diverse soliton solutions.
Family: 1
When .
We derive the periodic wave solution for (30) by adopting the Jacobi amplitude function as .
35
and by the relationship, , we follow
36
In the scenario where tends to 1, Eq (36) transforms into the shock wave solution for Eq (1) as indicated by
37
Family: 2
When .
We derive the periodic wave solution for (30) by adopting the Jacobi amplitude function as .
38
and
39
In the scenario where tends to 1, Eq (39) transforms into the singular soliton wave solution for Eq (1) as indicated by
40
Family: 3
When .
We derive the periodic wave solution for (30) by adopting the Jacobi amplitude function as .
41
and
42
In the scenario where tends to 0, Eq (42) transforms into the singular soliton wave solution for Eq (1) as indicated by
43
Likewise, as approaches 1, we obtain a singular soliton solution for Eq (1) given by
44
Family: 4
When .
We derive the periodic wave solution for (30) by adopting the Jacobi amplitude function as .
45
and
46
In the scenario where tends to 1, Eq (46) transforms into the optical bright soliton wave for Eq (1) as indicated by
47
Family: 5
When .
We derive the periodic wave solution for (30) by adopting the Jacobi amplitude function as .
48
and
49
In the scenario where tends to 1, Eq (49) transforms into optical bright soliton solution for Eq (1) as indicated by
50
Family: 6
When .
We derive the double periodic wave solution for (30) by adopting the Jacobi amplitude function as .
51
and
52
In the scenario where tends to 1, Eq (52) transforms into
53
Family: 7
When .
We derive the double periodic wave solution for (30) by adopting the Jacobi amplitude function as .
54
and
55
In the scenario where tends to 1, Eq (55) transforms into
56
Family: 8
When .
We derive the double periodic wave solution for (30) by adopting the Jacobi amplitude function as .
57
along with
58
In the scenario where tends to 1, Eq (58) transforms into
59
Family: 9
When .
We derive a rational solution for (30) by adopting the amplitude function as .
60
and
61
In the scenario where tends to 0, Eq (61) transforms into
62
Soliton solutions for TB model (1) using expansion method
The solution (8) assumes the following mathematical expression
63
We substitute Eq (63) into Eq (30), and then compare the polynomials of the type results in the following system
64
65
The following outcomes were acquired through the utilization of the Mathematica software
Set: 1, , , , , , , ,
66
where , and represent arbitrary constants and . By considering families following solution families are obtained
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
Set: 2, , , , , , , , , .
85
where , and represent arbitrary constants and . By considering families one can the following solutions can be obtained:
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
Set: 3, , , , , , , .
104
where, , and represent arbitrary constants, and . We consider families leading to
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
Set: 4, , , , , , , , , , .
123
where, , and represent arbitrary constants, and . By considering families leads to following results
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
Set: 5, , , , , , , , .
142
where, , and represent arbitrary constants, and . By considering families leads to
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
Analysis of the physical implications of the obtained results
This section provides a concise summary of the outcomes derived in the preceding sections. The theory of periodic and soliton solutions constitutes a fundamental and well-established domain in the modern theory of differential equations. These solutions play a crucial role in the analysis of dynamical systems and find applications across various fields, including mathematical biology, social sciences, and other nonlinear sciences, where phenomena are modeled with diverse parameters. Hence, it is essential to explore the conditions associated with these arbitrary parameters that give rise to periodic wave and soliton solutions. Graphical representations were used to elucidate the physical characteristics of the obtained solutions. In Fig. 1, the 3D and 2D plots illustrate the solution , featuring both real and imaginary components. This solution portrays a sub-picosecond shock wave within the intervals and , with the parameter values set as , , and all other arbitrary parameters set to unity. Figure 2 depicts the 3D and 2D plots of the solution , showing both real and imaginary aspects. These graphs illustrate a singular soliton solution within the intervals and , where the parameter values are specified as and , and all other arbitrary parameters are set to unity. Figure 3 displays the profiles of the solution , illustrating sub-picosecond singular wave solutions with ranges and . The parameter values were set as and , and all other arbitrary parameters were assigned a value of unity. Figure 4 illustrates the 3D and 2D plots of the solution , showing a sub-picosecond bright soliton solution over the spatial and temporal intervals and . The parameters are specified as and , and the remaining units. The plots in Fig. 5 correspond to the solution , depicting a double periodic wave solution within the intervals and . The parameters were set as and , and all other parameters were assigned a value of unity. Figure 6 illustrates the periodic wave solutions over the ranges and , derived from the solution . The parameters were set to , and all other arbitrary elements were set to unity.
Profile of periodic wave solution by setting parameters and .
Discussion and conclusions
This study explored subpicosecond optical soliton solutions within the TB model. By leveraging advanced techniques, specifically the gJEF method and the -expansion method, a diverse range of wave solutions, including sub-picosecond shock wave solitons, sub-picosecond optical bright and singular solitons, and double periodic waves, were systematically derived. Distinct from previous works by Yıldırım50 and Ghazala and Sayed51, our results introduce novel solution classes such as shock waves and double periodic wave solutions.
Furthermore, for the sake of novelty, a rigorous exploration of periodic wave solutions for the TB model (1) was undertaken. Nine periodic wave solutions, expressed in terms of Jacobi amplitude symbols and others, were obtained using the -expansion method. This novel contribution enriches our theoretical understanding of the TB model (1). The outcomes of this study underscore the necessity for a more intricate examination of the model. Future investigations may extend the TB equation relevant to birefringent fibers and Dense Wavelength Division Multiplexing (DWDM) technology, employing robust methodologies such as extended Kudryashov’s methodology, trial equation procedures, and Lie symmetry analysis. The comprehensive findings of these studies will be presented in subsequent publications.
Acknowledgements
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a large group Research Project under the grant number RGP2/55/46.
Author contributions
Writing original draft, Akhtar Hussain; Writing review and editing, Akhtar Hussain, Tarek F. Ibrahim, Arafa A. Dawood, and Faizah D Alanazi; Methodology, Akhtar Hussain, Tarek F. Ibrahim, and Ariana Abdul Rahimzai; Software, Akhtar Hussain; Supervision, Tarek F. Ibrahim and Ariana Abdul Rahimzai; Project administration, Ariana Abdul Rahimzai, and Tarek F. Ibrahim; Visualization, Akhtar Hussain, Waleed M. Osman, and Faizah D Alanazi; Conceptualization, Akhtar Hussain, and Ariana Abdul Rahimzai; Formal analysis, Arafa A. Dawood, Ariana Abdul Rahimzai, and Akhtar Hussain; Response to reviewers and revision; Akhtar Hussain, and Waleed M. Osman.
Funding
The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA for funding this research work through the project number “NBU-FFR-2025-1102-02”.
Data availability
All data generated or analyzed during this study are included in this published article.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
All data generated or analyzed during this study are included in this published article.
expansion method. Trigonometric, hyperbolic, exponential, polynomial, and rational functions are obtained. The physical dynamics of the obtained solutions confirmed the existence of known complex structures, such as shock waves, dark solitons, periodic waves, and singular periodic solutions. The simulations generated in Mathematica 11.3 are graphically presented to depict the nature of the acquired solutions. These results are novel and have not been reported previously in the literature.
-expansion technique, conservation laws, first integral technique, and traveling wave hypothesis43–45. Numerous46–48 other studies exist in the literature irrespective of conservation laws.
-expansion method55,56. The primary objective is to recover subpicosecond optical soliton solutions and ascertain the conditions that govern their existence. Additionally, the adopted methods led to the discovery of supplementary solutions, including shock waves, double periodic waves, and singular periodic solutions, facilitated by the reverse formulation of the constraints. A comprehensive analysis of the model’s intricacies is presented in subsequent sections of this article. None of the ansatz methods are so strong that they can deal with all types of solutions for each NLPDE. The generalized Jacobi elliptic function method does not apply to nonlinear problems/PDEs, where the product of the even and odd terms appears as a single term. This section covers the remaining cases. In addition, it is very difficult to deal with some classes of variable coefficient NLPDEs using both techniques.
, ensuring the presence of group velocity dispersion in the model. The profile of subpicosecond optical solitons is represented by the complex-valued function Q(x, t). The non-Kerr dispersion effect is counteracted by coefficient ‘
’ when 
. When the nonlinearity parameter takes the value of 
, the model aligns with the Kaup-Newell model. Conversely, when 
, the significance of the derivative quintic non-Kerr nonlinearity terms becomes pronounced in the transmission of extremely short pulses, characterized by widths around sub-10 fs, within highly nonlinear optical fibers.
denotes the solution of the nonlinear PDE (2). Using this transformation, we obtain
represent the soliton frequency, 
denotes the soliton wave, 
signifies the soliton phase, and c represents the speed of the wave. Then the nonlinear PDE (2) can be transformed into an ordinary differential equation (ODE) as follows
is necessary and the function 
satisfies the solution
and 
are parameters.
, 
, and 
, 
, and 
conform to the prescribed relationships
, the Jacobi elliptic function degenerate to the triangular functions,
, the Jacobi elliptic function degenerate to the hyperbolic functions,
and the hyperbolic functions for 
are detailed in Table 2.
.
(where 
) and 
(where 
) have yet to be determined. Function 
and 
,
and 
and 
,
and 
,
,
,
,
and 
,
,
,
,
,
,
,
and 
. A system of algebraic equations was derived by setting each coefficient to zero. When solved using Mathematica software, these equations allow for the determination of the values of 
, 
, 
, 
, and 
.
, 
, ..., 
, so one can get
, the Eq (5) suggests
, (34) ascertains diverse soliton solutions.
.
.
, we follow
tends to 1, Eq (36) transforms into the shock wave solution for Eq (1) as indicated by
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
results in the following system
, 
, 
, 
, 
, 
, 
, 
, 
, and 
. By considering families 
following solution families are obtained
, 
, 
, 
, 
, 
, 
, 
.
, 
, 
, 
, 
.
, 
, 
, 
, 
, 
, 
.
, 
, 
, 
.
, featuring both real and imaginary components. This solution portrays a sub-picosecond shock wave within the intervals 
and 
, with the parameter values set as 
, 
, showing both real and imaginary aspects. These graphs illustrate a singular soliton solution within the intervals 
and 
, where the parameter values are specified as 
, illustrating sub-picosecond singular wave solutions with ranges 
, showing a sub-picosecond bright soliton solution over the spatial and temporal intervals 
, depicting a double periodic wave solution within the intervals 
and 
, derived from the solution 
. The parameters were set to 
, and all other arbitrary elements were set to unity.
and 
and 
.
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