Abstract
This research explores the analytical soliton solutions of the dimensionless time-dependent paraxial equation (DTPE). We applied the improved Sardar sub-equation and improved Riccati equation analytical techniques to develop numerous soliton solutions for the DTPE, providing insights into the behavior of optical solitons in nonlinear evolution equations. The two techniques are applied for the first time to generate vigorous solutions to the DTPE. Numerous types of exponential, trigonometric, hyperbolic, dark, periodic, anti-kink, peakon, kink, and bright soliton solutions are obtained for DTPE. Although constructing an effective scheme to solve the DTPE has been our primary aim. Moreover, the obtained analytical solutions offer a useful foundation for comprehending the complex dynamics of optical solitons in nonlinear evolution equations, enabling enhancements in different applications that include signal processing, optical communication, beam shaping, Gaussian beams, lenses and mirrors, optical fibers, and predicting beam behaviors.
Keywords: Nonlinear paraxial wave equation, Improved Sardar sub-equation method, Improved Riccati equation method, Soliton solutions, Symbolic computation
Subject terms: Mathematics and computing, Optics and photonics, Physics
Introduction
The investigation of nonlinear optical waves has arisen as a crucial area in nonlinear optics, portrayed by the intriguing interface between dispersion and nonlinearity in wave propagation1–3. Specifically, the dimensionless time-dependent paraxial equation (DTPE) functions as an essential foundation for studying the interface of optical beams in different media4,5. Studying this equation has important applications in nonlinear optics6–8, optical fiber communication, optical simulation software, beam shaping, Gaussian beams9, lenses and mirrors, optical fibers, and predicting beam behavior10,11. The DTPE is mathematically represented as follows29:
 |
1 |
Where 
, 
is a symbol representing dispersion, 
is diffraction, and 
Kerr nonlinearity. This equation portrays the adaptation of light beams as they navigate via nonlinear materials, offering an understanding of phenomena like mode locking, self-focusing, and pulse compression12.
Moreover, optical solitons, as a specific type of solitary wave solution for this equation, warrant important attention owing to their capability to preserve a confined form at the time of propagating continuously without deformation13. Optical solitons have applications in various fields of science and engineering14–18. Various mathematical approaches have been applied to obtain the analytical solutions to the DTPE, such as auxiliary equation approach19, the unified equation method20, direct algebraic approach21, the extended modified rational function method22, the extended simple equation method28, and improved F-expansion approach23. Many other analytical and computational techniques have been used to investigate the various problems in the existing literature24–27,29. Trigonometric, dark, periodic, kink, and bright solutions were found as a result of these analytical methods applied to the DTPE30–32. However, the literature of analytical methods for solving the DTPE looks scant33–35, indicating the need to apply more powerful techniques to derive robust solutions for DTPE36–38. This study will attempt to give an extensive analytical investigation of optical soliton solutions to the DTPE. We will focus on promoting the comprehension of solitonic behavior in nonlinear optics and investigate the possible applications in beam propagation, optical design, and advanced laser systems39,40. This research aims to treat the research gaps in the literature by deducing new solitary wave solutions to the DTPE using the improved Sardar sub-equation method (ISSEM) and the improved Riccati equation method (IRM). These methods have an exceptional advantage in deriving numerous solutions in the form of different wave functions to the nonlinear evolution equations; see41–43 on the advantages and exceptionalities of applying the identified methods.
This paper is organized as follows: In Section 2, the description of the methods is discussed. In Section 3, the solutions of the DTPE are presented using the identified methods. Results and discussion of the obtained solutions are explained graphically in Section 4, and the conclusion is given in Section 5.
Description of the methods
This section will present and interpret the methods associated with solving Eq.(1) using the ISSEM and the IRM. In the general case, assume that given any nonlinear partial differential equation (NPDE) of the form44,45:
 |
2 |
the function 
is polynomial in 
, and its partial derivatives. The two methods transformed the NPDE to a nonlinear ordinary differential equation (NODE) using any suitable wave transformation, for example,
 |
3 |
where 
, 
. Now, plugging the above transformation and its derivatives into Eq.(2), we get
 |
4 |
where 
is a polynomial function in 
and its derivatives 
, 
, 
. The two techniques differ in how to derive the solution of the NODE. The details in each technique are as follows:
The ISSEM
This method assumes the solution of the NODE Eq. (4) to be
 |
5 |
where the constant parameters 
, for 
are to be determined. The value of 
can be secured by balancing the highest-order nonlinear and the highest-order derivative terms in Eq. (4). Whereas the polynomial function 
is any solution that satisfies
 |
6 |
The existence of real and regular solutions of Eq.(6) depends on the constraints conditions on 
, 
and 
as follows:
For
and 
, the rational form solution is 
-
2.
For
and 
, the exponential form solutions are:
-
3.
The trigonometric hyperbolic solutions are as follows:
(i) For 
, 
and 
we have:
(ii )For 
, 
and 
, we have:
-
4.
The trigonometric form solutions are as follows:
(i) For 
, 
and 
we have:
(ii) For 
, 
and 
, we have:
See46, for more details.
Where 
, 
, and 
are constants to be determined44. By substituting Eq.(5) and its derivatives into Eq.(4), then equating the coefficients of 
to zero and yielding a system of simple algebraic equations. Solving the generated equations to gain the values of the constants 
and 
that satisfy the NODE Eq.(4), and the NPDE Eq.(2) using each of the solutions of Eq.(6). Eq.(6) has various rational, exponential, hyperbolic, and trigonometric function solutions44.
The IRM
The IRM assumes the solution of the NODE Eq.(4) to be
 |
7 |
Where the constant parameters 
, for 
. The value of 
can be obtained by a homogeneous balancing procedure, and the polynomial function 
is a solution that satisfies the generalized Riccati equation45.
 |
8 |
The constraint conditions on 
, 
and 
determine the nature of the solutions (real, hyperbolic or periodic) of Eq.(8). The solutions of the auxiliary equation Eq.(8) are classified based on the discriminant 
The following conditions
For
, and 
the rational form solution is 
-
2.
For
, the exponential form solutions are:
-
3.
For
, 
or 
, with non-zero real constants m and n, the trigonometric hyperbolic form solutions are the following:
-
4.
For
, 
or 
, with 
, the trigonometric form solutions are the following:
See46, for more details.
Where 
, 
, and 
are constants to be determined. By substituting Eq.(5) and its derivatives into Eq.(4), then equating the coefficients of 
to zero and yielding a system of algebraic equations. Solving the generated equations to obtain the solution of the constants 
and 
that satisfy the NODE Eq. (4) and the NPDE Eq.(2) using each of the solutions of Eq.(8). The Eq.(8) has various families of solutions45.
To verify that the result 
indeed satisfies the reduced ODE Eq.(4), we substitute 
and its derivatives back into the reduced ODE Eq.(4) and check if it holds true.
The solutions of the DTPE
This section will apply the procedures highlighted in section 2 to derive the solutions of Eq. (1). Starting with transforming Eq.(1) into NODE by plugging the transformation Eq.(9) and its derivatives into Eq.(1).
 |
9 |
where p is a velocity, 
a frequency, a a phase constant, 
a wave number, 
, phase component, 
an amplitude component, 
a propagation coefficient along y-axis, 
is a transformed variable, and P(x, y, t) the pulse shape of the soliton.
 |
 |
 |
 |
 |
 |
Substituting the above transformation and its derivatives into Eq. (1), we get:
 |
10 |
 |
11 |
Separate the real and imaginary parts; we get the real part of Eq.(10) as:
 |
12 |
Collecting the like terms to have
 |
13 |
Also, we get the imaginary part of Eq.(10) as:
 |
14 |
thus, we get:
 |
15 |
Integrating Eq.(15) twice, we get:
 |
16 |
 |
17 |
The implementation of ISSEM to the DTPE
In this part, we will utilize ISSEM to obtain the solitary wave solutions to Eq.(13). Using the homogeneous balancing procedure between the highest-order derivative term 
with the highest-power nonlinear term 
to get,
 |
18 |
which yields 
. Now, plugging 
into Eq.(5), we get:
 |
19 |
as the solution of Eq.(13), where 
and 
are to be determined. Plugging Eq.(19) and its derivatives together with Eq.(6) into Eq.(13) and equating all the coefficients of 
, 
, 
and 
to zero, we get:
 |
20 |
Solving the above system of equations with the help of Maple 18, we get the following solutions:
 |
21 |
Case I:
For 
and 
, the exponential form solutions are:
 |
 |
Case II:
(i) For 
, 
and 
, we have:
 |
 |
(ii) For 
, 
and 
, we have:
 |
 |
 |
 |
 |
Case III:
(i) For 
, 
and 
, we have:
 |
 |
(ii) For 
, 
and 
, we have:
 |
 |
 |
 |
 |
From Eq. (9), we can get the solutions of Eq. (1) by substituting the values of 
above into Eq. (9) as follows:
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
The implementation of IRM to the DTPE
In this part, we will apply IRM to obtain the solitary wave solutions to Eq.(13). Using the homogeneous balancing procedure, in the same way as for the ISSEM, we get 
. Plugging 
into Eq.(7), we obtain:
 |
22 |
be the solution of the generalized Riccati equation, Eq.(8). Plugging Eq.(22) and its derivatives together with Eq.(8) into Eq.(13) and equating all the coefficients of 
, 
, 
and 
to zero, we get:
 |
23 |
Solving the system of equations with Maple 18, we obtain the following results:
 |
24 |
Case I:
For 
, 
(or 
), the trigonometric hyperbolic form solutions are as follows:
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
Case II:
For 
, 
(or 
), the trigonometric form solutions are as follows:
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
From Eq.(9), we can get the solutions of Eq.(1) by substituting the values of 
above into Eq.(9) as follows:
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
Where
,
, 
, and C is the integration constant.
Results and discussion
In this section, the obtained results for the DTPE are displayed graphically; we have successfully produced an enormous number of soliton structures. The graphical solutions of some of the obtained solutions are shown in Figures 1–12. These cover the 3D and 2D of the absolute, real, and imaginary graphs for various solutions. In the investigation carried out by the author of29, 15 solutions were recovered, while in our study, we recovered 20 novel solutions using IRM, and 16 solutions were also obtained by utilizing ISSEM. From the result obtained, we can observe that the present study supersedes the one carried out by29 by producing more soliton solutions. Some of the recovered solitary structures in Fig.1 and Fig. 2 are the 3D and 2D rogue wave solitons of 
, Re(
) and Im(
) at fixed t of ISSEM, with 
, 
, 
, 
, 
and 
, Fig. 3 and Fig. 4 displayed the 3D and 2D bright soliton of the solution 
, periodic solitons of Re
) and Im(
) at fixed t of ISSEM, with 
, 
, 
, 
, 
and 
. Bright solitons are important in fiber optics as they sustain their form across great distances by adjusting dispersion and nonlinearity. Bright solitons function as standard transmission medium for wavelength-division transmission integration systems because of their stable interactions. Their regulatable properties make them crucial for advancing emerging optical communication networks and high-speed pulse lasers. While periodic soliton exhibits the stability and repeated features usual to recurring profiles in nonlinear science. The solitons offer a robust solution to numerous problems in contemporary communication networks. Their unique capability to maintain transmission quality and reduce loss makes them crucial in modern transmission technology. Fig. 5 and Fig. 6 are 3D and 2D structures of rogue waves of the solution 
, Re(
) and Im(
) at fixed t of ISSEM, with 
, 
, 
, 
, 
and 
. Fig. 7 and Fig. 8 presented a dark soliton of the solution 
, periodic solitons of Re(
) and Im(
) of IRM at fixed t, with 
, 
, 
, 
, 
and 
, Dark solitons being indispensable in optical fiber for their outstanding stability as intensity sags on a continuous framework, facilitating robust extensive wave propagation. Their distinctive properties enable operation in usual dispersion systems, expanding functional fiber bandwidth. While Fig. 9 and Fig. 10 represented the kink soliton of 
, periodic solitons of Re(
) and Im(
) at fixed t of IRM, with 
, 
, 
, 
, 
and 
, and lastly, Fig. 11 and Fig. 12 showed the evolution of the rogue waves of the solutions of 
, Re(
) and Im(
) at fixed t of IRM, with 
, 
, 
, 
, 
and 
.
Fig. 1.
The rogue wave solitons of 
, Re(
) and Im(
) at fixed t of ISSEM, with 
, 
, 
, 
, 
and 
.
Fig. 12.
The rogue wave solitons of 
, Re(
) and Im(
) at 
, 0.2 and 0.4 of IRM, with 
, 
, 
, 
, 
and 
.
Fig. 2.
The rogue wave solitons of 
, Re(
) and Im(
) at 
, 0.2 and 0.4 of ISSEM, with 
, 
, 
, 
, 
and 
.
Fig. 3.
The bright of 
and periodic solitons of Re
) and Im(
) at fixed t of ISSEM, with 
, 
, 
, 
, 
and 
.
Fig. 4.
The bright soliton of 
and periodic solitons of Re(
) and Im(
) at 
, 0.2 and 0.4 of ISSEM, with 
, 
, 
, 
, 
and 
.
Fig. 5.
The rogue wave solitons of 
, Re(
) and Im(
) at fixed t of ISSEM, with 
, 
, 
, 
, 
and 
.
Fig. 6.
The rogue wave solitons of 
, Re(
) and Im(
) at 
, 0.2 and 0.4 of ISSEM, with 
, 
, 
, 
, 
and 
.
Fig. 7.
The dark soliton of 
and periodic solitons of Re(
) and Im(
) at a fixed t of IRM, with 
, 
, 
, 
, 
and 
.
Fig. 8.
The dark soliton of 
and periodic soliton of Re(
) and Im(
) at 
, 0.2 and 0.4 of IRM, with 
, 
, 
, 
, 
and 
.
Fig. 9.
The kink soliton of 
and periodic solitons of Re(
) and Im(
) at a fixed t of IRM, with 
, 
, 
, 
, 
and 
.
Fig. 10.
The kink soliton of 
and periodic solitons of Re(
) and Im(
) at 
, 0.2 and 0.4 of IRM, with 
, 
, 
, 
, 
and 
.
Fig. 11.
The rogue wave solitons of 
, Re(
) and Im(
) at fixed t of IRM, with 
, 
, 
, 
, 
and 
.
Conclusion
This research offered significant understanding into the behavior of optical systems and offers a comprehensive structure for solving the DTPE utilizing sophisticated mathematical techniques, that is, the ISSM and IRM. The derived solutions are in the form of exponential rational functions, trigonometric functions, hyperbolic functions, complex hyperbolic functions, and complex trigonometric function solutions. Firstly, our proposed methods utilize a set of parameters 
, 
and 
and yields a different structure and novel formulation different from the one obtained in the existing literature. In addition to setting the foundation for further research and valuable applications in various areas, it also enhanced the science of nonlinear optics. These research results may have applications in plasma waves, signal transmission, coastal engineering, optical fiber, hydromagnetic waves, quantum mechanics, and nonlinear optics, underlining the capacity and efficiency of the applied techniques in investigating nonlinear equations for contemporary science and engineering. The investigation further forms a powerful bedrock for subsequent investigations and the profiling of soliton solutions in the form of dark, rogue, periodic, kink, and bright wave structures. In brief, our obtained results, which are the major success in this research, are considerably unique from those of the compared technique in29. Additionally, our proposed techniques take advantage of a quite easy but effective way to achieve analytical results, which is more efficient than some current techniques. Furthermore, this study importantly contributes to the enhancement of understanding in the area of DTPE and sets the stage for upcoming advancements and applications.
Acknowledgements
The authors extend their appreciation to Princess Nourah bint Abdualrahman University Researchers Supporting Project number (PNURSP2026R184), Princess Nourah bint Abdualrahman University, Riyadh, Saudi Arabia.
Author contributions
ISI: Writing-Original draft preparation, Methodology. JS: Supervision, Conceptualization. MI: Conceptualization, Methodology. SS: Investigation, Writing-Original draft preparation. WGA: Software, Data curation. SG: Visualization, Validation. NMN: Formal analysis, Writing-Reviewing & Editing.
Data availability
All data that support the findings of this study are included in the article.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Mujahid Iqbal, Email: mujahidiqbal399@gmail.com.
Nazar Mohammad Nazar, Email: n.moh@nu.edu.af.
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Data Availability Statement
All data that support the findings of this study are included in the article.