Explicit optical solitons of a perturbed Biswas-Milovic equation having parabolic-law nonlinearity and spatio-temporal dispersion

Objective – This paper deals with a new variant of the Biswas-Milovic equation, referred to as the perturbed Biswas-Milovic equation with parabolic-law nonlinearity in spatio-temporal dispersion. To our best knowledge, the considered equation which models the pulse propagation in optical ﬁber is studied for the ﬁrst time, and the abundant optical solitons are successfully obtained utilizing the auxiliary equation method. Methods – Utilizing a wave transformation technique on the considered Biswas-Milovic equation, and by distinguishing its real and imaginary components, we have been able to restructure the considered equation into a set of nonlinear ordinary diﬀerential equations. The solutions for these ordinary diﬀerential equations, suggested by the auxiliary equation method, include certain undetermined parameters. These solutions are then incorporated into the nonlinear ordinary diﬀerential equation, leading to the formation of an algebraic equation system by collecting like terms of the unknown function and setting their co-eﬃcients to zero. The undetermined parameters, and consequently the solutions to the Biswas-Milovic equation, are derived by resolving this system. Results – 3D, 2D, and contour graphs of the solution functions are plotted and interpreted to understand the physical behavior of the model. Furthermore, we also investigate the impact of the parameters such as the spatio-temporal dispersion and the parabolic nonlinearity on the behavior of the soliton. Conclusion – The new model and ﬁndings may contribute to the understanding and characterization of the nonlinear behavior of pulse propagation in optical ﬁbers, which is crucial for the development of optical communication systems.


Introduction
Nonlinear partial differential equations (NLPDEs) are widely utilized to model physical phenomena in diverse areas such as fluid dynamics [1], electromagnetism [2], quantum mechanics [3], material science [4], and engineering due to their ability to model close to reality.One of the areas where NLPDEs are widely used is in nonlinear optics [5,6].Because the wave propagation in optical fibers can be modeled well with the help of NLPDEs, especially, the nonlinear Schrödinger equations (NLSEs).Numerous models have been developed in nonlinear optics such as Davey-Stewartson [7], Biswas-Arshed [8], Ginzburg-Landau [9], and Biswas-Milovic [10][11][12].The field of nonlinear optics is characterized by various concepts such as dispersion (chromatic [13], third-order [14], fourth-order [15], higher-order [16], and spatio-temporal [17], etc.), nonlinearity (Kerr [18], parabolic [19], cubic [20], quartic [21], quintic [22], and sextic [23], etc.), refractive index (Kudryashov's refractive index [24,25]), and perturbation [26,27].These concepts have led to the development of new forms of the many classical equations in nonlinear optics [28][29][30].Nonlinear optics has gained importance in the last two decades due to its widespread use in communication, electronics, the internet, social media, computers, and data transfer [31,32].The wave behavior in optical fibers is complex and nonlinear.Solitons that can travel long distances in a medium without any significant dispersion or distortion, play a vital role in the transmission of light signals through an optical fiber.So, soliton solutions of NLSEs are critical in exploring the nonlinear behavior of pulse propagation in optical fiber.Despite the inherent difficulty of solving NLSEs, recent advancements in computer algebraic systems such as Mathematica [33], Maple [34], and Matlab [35] have made it possible to find solutions to NLSEs that were previously considered unsolvable analytically.In recent times, since solving NLSEs has attracted great interest of researchers, a wide range of analytical methods have been put forward in the literature to solve them.The analytical methods proposed encompass a broad range of approaches, including but not limited to the unified Riccati equation expansion [36,37], F-expansion [38], Fan's method [39], Kudryashov methods [12,[40][41][42][43], exp-function [44], G ′ /G-expansion method [45], sine-Gordon expansion method [46], extended sinh-Gordon equation [47,48], the modified extended tanh function [49], and many more [50][51][52].The dimensionless form of the Biswas-Milovic equation (BME) was introduced as follows [10]: in which q = q(x, t) is a complex-valued function.β is a real constant, n ⩾ 1 is a nonlinearity parameter that transforms the NLSE to its generalized form.The real-valued algebraic function F needs to ensure the smoothness of the complex function When the complex plane is considered C as a two-dimensional real space R 2 , F (|q| 2 ) q is expected to be continuously differentiable for k times such that [10]: The Biswas-Milovic equation in the presence of spatio-temporal dispersion (STD) is defined by [11]: in which q = q(x, t) indicates a complex-valued function, α, β shows the coefficient of spatio-temporal effect, the group velocity dispersion, respectively.In [11], using the inverse scattering transform and the Darboux transformation methods, the authors studied exact solutions of Biswas-Milovic equation with spatio-temporal dispersion and diverse nonlinearities such as Kerr law, dual power law, and triple power-law.Supposing F (q) = c 1 q + c 2 q 2 , we present the perturbed Biswas-Milovic equation with parabolic-law nonlinearity in the presence of spatio-temporal dispersion as follows: ) in which q = q(x, t) denotes a complex field envelope.Here, α is the coefficient of spatiotemporal dispersion, β is the coefficient of group velocity dispersion, c 1 and c 2 represent the coefficients of cubic and quintic power-law nonlinearities of self-phase modulation, respectively.λ denotes the coefficients of self-steepening while τ and µ are the coefficients of nonlinear terms.The BME is one of the well-known NLPDEs that arise in the study of fluid dynamics, heat transfer, nonlinear optics, etc.Many researchers have already studied soliton solutions to the diverse forms of BME such as (2+1) and (3+1) dimensional BME [53], BME having spatio-temporal dispersion and parabolic law [12], BME with dual-power law nonlinearity and multiplicative white noise [54], BME with Kudryashov's law of refractive index [55].However, to our best knowledge, the perturbed Biswas-Milovic equation with parabolic-law nonlinearity in the presence of STD has not been studied before.So, the primary goal of this paper is to present a new form of BME and contribute to the literature by extracting abundant and different kinds of soliton solutions of the considered equation.Specifically, we aim to apply the auxiliary method to obtain various types of solitons for this equation.The following parts of the paper are structured in this way: A wave transformation is applied to the NLPDE in section 2. In section 3, we briefly give a review of the auxiliary equation method.And then, the auxiliary equation approach is utilized to get the soliton solutions of the considered BME in section 4. In section 5, we produce some simulations to investigate the behavior of the obtained solutions.Lastly, in Section 6, various concluding remarks and suggestions for future works are provided.

Wave Transformation
The following transformation is considered for the model in eq. ( 4): in which v, ψ, κ, ω, and ϕ are reals to be found such that ψ denotes the component of phase, κ indicates the frequency, ω is the wave number, ϕ is the center of phase, and v represents the velocity.By separating the real and imaginary parts and equaling them to zero, the nonlinear ordinary differential equations (NLODEs) are attained, respectively: and where Q = Q(η) and the superscripts ′ and ′′ represent dQ(η) dη and d 2 Q(η) dη 2 , respectively.Equation (7) yields the following constraint conditions: where αn ̸ = 0.In eq. ( 6), balancing the terms QQ ′′ and Q 6 , we calculate the balance number as N = 1 2 .So, let us consider the following transformation: Replacing eqs.( 8) and ( 9) into eq.( 6), the following equation is found: where U = U (η) and (11)

Auxiliary Equation Method
In accordance with the approach in [56], let us presume that eq. ( 10) is of the solutions below: where the positive integer N stands for the balance number.Ω 0 , Ω 1 , . . ., Ω N are real values that need to be found, and S(η) represents the solution of the ordinary differential equation: where p i (i = 1, 2, 3) are reals and the solutions of eq. ( 13) are presented as: where ∆ = p 2 2 − 4p 1 p 3 , and ε = ±1.

Application
Applying the principle of homogeneous balance to the terms UU ′′ and U 4 in eq. ( 10) yields N = 1.So, eq. ( 12) can be re-written as follows: Upon inserting eq. ( 15) and its corresponding derivatives in eq. ( 10) and equaling the same powers of S(η) into zero, we find: Solving the system in eq. ( 16), one gets: Considering the sets above and eqs.( 14), ( 15) and ( 18), together, one derives the solutions of the NLODEs in eq.(10).Using the wave transformation in eq. ( 5) and the transformation in eq. ( 9), the solutions for eq.( 4) can be derived as follows: where

Results
In this study, the perturbed Biswas-Milovic equation with parabolic-law nonlinearity in the presence of STD was considered.We applied the auxiliary equation technique to obtain soliton solutions of the considered BME.Besides, we have investigated the impacts of the parameters in the considered equation.All computations were carried out using Mathematica 13 [33].The obtained soliton solutions were presented in some figures to illustrate their properties via Matlab [35].These figures provide important insights into the dynamics of the solution functions.
In fig. 1, the some graphs of q − 1 (x, t) in eq. ( 20) are plotted for Set 1 1b.The two-dimensional graphs of Re q − 1 (x, 3) (red line), Im q − 1 (x, 3) (green line) and |q     In fig.2, some plots of q − 11 (x, t) in eq. ( 30) with specific parameter values, namely, Set 1 contains three sub-figures.Figure 2a illustrates a three-dimensional plot of the modulus square of q − 11 (x, t) while fig.2b shows a two-dimensional contour plot of the modulus square of q − 11 (x, t).In fig.2c, the blue and red lines indicate the imaginary and real parts of q − 11 (x, 3), respectively.Additionally, the figure includes blue-dotted, blue-dashed, and blue-solid lines that represent the two-dimensional plots of |q      In fig.4, the used parameters are (a) The effect of c 2 .
(a) The effect of p 1 .
Figure 6: The impact of p 1 and p 2 on the behavior of q − 1 (x, t) in eq. ( 20) (bright soliton).
In fig. 7, the effect of the parameter p 3 in the auxiliary equation in eq. ( 13) is examined, respectively.The parameters are selected as n As a result, the analysis presented can provide insights into the dynamics of the BME and highlights the importance of various parameter values in shaping the behavior of the solution.

Conclusion
In this study, the soliton solutions of the perturbed Biswas-Milovic equation with paraboliclaw nonlinearity in the presence of STD which explains the pulse propagation in optical fiber were successfully obtained using the auxiliary approach.We get the bright, kink, and singular solutions.The graphs of obtained solutions have been shown using Matlab to comprehend the solitons' behavior.Furthermore, we have investigated the impact of different parameters in the considered equation on the behavior of the solitons, such as the amplitude.It is seen that the amplitude of the waves is affected by the values of parameters in the equation, such as c 2 , α, p 1 , p 2 , and p 3 .The obtained results reveal that the used method is a useful and efficient technique for extracting abundance and diverse kinds of solitons.As a result, this work provides a comprehensive analysis of the soliton solutions of the new form of BME and demonstrates the effectiveness of the auxiliary equation method.
The new model and its soliton obtained may help to explore the nonlinear behavior in pulse propagation in optical fibers, which is crucial for the development of optical communication systems.

Declarations Ethical Approval
This study was carried out in accordance with the recommendations of IOP Publishing's statement.This research did not contain any studies involving animal or human participants, nor did it take place on any private or protected areas.The Corresponding Author declares that this manuscript is original, has not been published before, and is not currently being considered for publication elsewhere.The Corresponding Author confirms that the manuscript has been read and approved by all the named authors and there are no other persons who satisfied the criteria for authorship but are not listed.The Corresponding Author further confirms that the article includes just one author who is Melih Cinar.
− 1 (x, t f )| 2 for various values of t f = 1 (blue-solid line), t f = 3 (blue-dashed line), and t f = 5 (blue-dotted line) are shown in fig.1c.Based on the figures of the solution with the chosen parameter values, |q − 1 (x, t)| 2 represents a bright soliton which moves towards the right on the x-axis, as can be observed from fig. 1c.
− 11 (x, t f )| 2 for t f = 1, 3, 5, respectively.Based on the chosen parameters, |q − 11 (x, t)| 2 indicates the presence of a kink soliton, which can be observed moving towards the right direction along the x− axis in fig.2c as t f increases.
and Set 1 − .Figure4apresents a three-dimensional graph, indicating the square of the magnitude of q − 14 (x, t), while fig.4bdisplays a two-dimensional contour graph.It has a singular solution behavior.Furthermore, fig.4cexhibits a combination of two-dimensional graphs, i.e. |q − 14 (x, t f )| 2 at fixed time values t f , along with the real and imaginary components of q − 14 (x, 3).In the last subfigure, blue dotted, blue dashed, and blue solid lines represent the graphs of |q − 14 (x, t f )| 2 to time points t f = 1, t f = 3, and t f = 5, respectively.
The effect of α.