Search

Purpose: The purpose of the article is to conduct a mathematical and theoretical analysis of soliton solutions for a specific nonlinear evolution equation known as the (2 + 1)-dimensional Zoomeron equation. Solitons are solitary wave solutions that maintain their shape and propagate without changing form in certain nonlinear wave equations. The Zoomeron equation appears to be a special model in this context and is associated with other types of solitons, such as Boomeron and Trappon solitons. In this work, the authors employ two mathematical methods, the modified F-expansion approach with the Riccati equation and the modified generalized Kudryashov's methods, to derive various types of soliton solutions. These solutions include kink solitons, dark solitons, bright solitons, singular solitons, periodic singular solitons and rational solitons. The authors also present these solutions in different dimensions, including two-dimensional, three-dimensional and contour graphics, which can help visualize and understand the behavior of these solitons in the context of the Zoomeron equation. The primary goal of this article is to contribute to the understanding of soliton solutions in the context of the (2 + 1)-dimensional Zoomeron equation, and it serves as a mathematical and theoretical exploration of the properties and characteristics of these solitons in this specific nonlinear wave equation. Design/methodology/approach: The article's methodology involves applying specialized mathematical techniques to analyze and derive soliton solutions for the (2 + 1)-dimensional Zoomeron equation and then presenting these solutions graphically. The overall goal is to contribute to the understanding of soliton behavior in this specific nonlinear equation and potentially uncover new insights or applications of these soliton solutions. Findings: As for the findings of the article, they can be summarized as follows: The article provides a systematic exploration of the (2 + 1)-dimensional Zoomeron equation and its soliton solutions, which include different types of solitons. The key findings of the article are likely to include the derivation of exact mathematical expressions that describe these solitons and the successful visualization of these solutions. These findings contribute to a better understanding of solitons in this specific nonlinear wave equation, potentially shedding light on their behavior and applications within the context of the Zoomeron equation. Originality/value: The originality of this article is rooted in its exploration of soliton solutions within the (2 + 1)-dimensional Zoomeron equation, its application of specialized mathematical methods and its successful presentation of various soliton types through graphical representations. This research adds to the understanding of solitons in this specific nonlinear equation and potentially offers new insights and applications in this field. [ABSTRACT FROM AUTHOR]

In this work, we take into account the (2+1)-Davey Stewartson equation (DSE) and the (2+1)-complex coupled Maccari system (CCMS) and their analytical solutions. Besides, we tackle the role of the problem parameters on the soliton behavior produced by the presented DSE. Exact traveling wave solutions are highly useful in numerical and analytical theories for such equations. While numerical methods are widely used, improving analytical approaches for obtaining analytical solutions is necessary for a deeper understanding of dynamics. This study marks a significant milestone by implementing the efficient analytical approach, enhanced modified extended tanh expansion method, for the first time to the (2+1)-DSE and (2+1)-CCMS equations, thereby making a notable contribution to the existing literature. We have shown that features of the soliton solutions can represent the spread of propagation on the wavefronts and show a reasonable dependency on parameter values. Some of the solutions discovered in three- and two-dimensional arrangements can also be described in graphic representations of their behavior. With the help of the graphical depictions, bright, singular, and periodic singular soliton characters for the (2 + 1) -DSE and singular, dark, bright, and periodic singular soliton characters for the (2 + 1) -CCMS are acquired. The results show that the utilized analytical technique is easily applicable, efficient, reliable, robust, and categorical when it comes to finding analytical solutions for different nonlinear models. Moreover, the problem parameters and the coupling coefficients have significant influences on the behavior of the solitons of the DSE, and this examination is studied for the first time in this article. [ABSTRACT FROM AUTHOR]

This study purposes to extract some fractional analytical solutions of the Peyrard-Bishop-Deoxyribo-Nucleic-Acid (β -PBDNA) dynamic model with the beta-derivative by the unified Riccati equation expansion method (UREEM). Furthermore, we examine the role of various parameters of the fractional model on the soliton dynamic. The research focuses on computational biophysics and materials science, examining the impact of various parameters on the fractional model. This paper contributes to understanding soliton solutions and the β -PBDNA dynamic model, demonstrating the applicability of the UREEM method to various fractional models. Some soliton solutions of the model are successfully generated by applying the UREEM. Implementing the UREEM, we take a fractional wave transformation to convert the model into a nonlinear ordinary differential equation. So, a linear equation system is generated. After the system is solved, the soliton solutions are gained for the appropriate solution sets. Finally, 3D, 2D and contour graphs of diverse solutions are depicted at suitable values of parameters. In addition, this paper presents 3D, 2D and contour graphs of various solutions with suitable parameter values. The results are beneficial for interpreting the model in future work and confirm that UREEM is effectively applicable to diverse fractional models, coupled with a comprehensive graphical analysis of how different parameters influence these solutions. [ABSTRACT FROM AUTHOR]

This study aims to obtain the optical soliton solutions of the stochastic perturbed Radhakrishnan–Kundu–Lakshmanan equation which models the optical solitons in optical waveguides such as silica fibers with Kerr law in the presence of chromatic and third-order dispersions by multiplicative white noise in Itô calculus. The study constitutes one of the works aimed at adapting the model in accordance with the importance of the noise effect in nonlinear optics. In the first step, the nonlinear ordinary differential form of the investigated problem has been obtained by using an appropriate complex wave transformation. The enhanced Kudryashov technique, which is the combination of Kudryashov and the new Kudryashov methods, is applied to the nonlinear ordinary differential form. As the result, bright, dark and singular stochastic soliton solutions of the analyzed problem have been carried out by assigning appropriate parameter values and the impact of noise effect on the soliton behavior has been investigated. Lastly, the graphical representations and required comments have been presented in the related sections. [ABSTRACT FROM AUTHOR]

In this article, we are interested in two principal topics. First, the bright optical soliton solutions of the third-order (1+1)-nonlinear Schrödinger equation including power law nonlinearity with inter-modal and spatio-temporal dispersions are perused by taking advantage of the new Kudryashov method. Second, the impacts of power law nonlinearity parameters on soliton attitude are investigated for acquired bright soliton form. With the proposed technique, the bright optical soliton solution is acquired, and 3D, contour, and 2D plots are depicted. Then, the impact of power law nonlinearity parameters on the soliton attitude has been successfully demonstrated. As is clear from this perusal power law parameters have an important impact on the soliton attitude, and this impact alters based on the soliton form. As regards our investigation, this form of the equation has not been studied with the power law nonlinearity in the absence of the chromatic dispersion for nonlinear models and the proposed method has not been applied the introduced equation before. It is expected that the consequences which are acquired in this study will shed light on the studies in this field. [ABSTRACT FROM AUTHOR]

The topic of dispersive optical solitons is one of the essential areas of telecommunications engineering. This paper presents the nonlinear Schrödinger equation (NLSE) with the cubic-quintic-septic (CQS) law which includes higher-order dispersion terms. This equation examines the dynamics of dispersive optical solitons propagation in nonlinear optical fibers. The presented model is significant in mathematics and theoretical physics sciences because long-distance optical fibers are used in optical communication utilized for transoceanic and intercontinental data transmission. First, the considered equation is transformed into a nonlinear ordinary differential equation (NLODE) using the traveling wave transformation. Then, two powerful techniques, Sinh-Gordon expansion approach and the auxiliary equation method (AEM), based on the homogeneous balance rule, are operated to produce analytical solutions for the NLSE with CQS law. Thus, bright, dark, singular, and periodic-singular soliton solutions are acquired. We demonstrate the three-dimensional (3D) and two-dimensional (2D) representations to interpret the physical behavior of some derived solutions. Besides, we investigate the effects of some parameters in the model with graphical illustrations. Therefore, it can be noted that the obtained soliton solutions have a significant role in soliton dynamics. [ABSTRACT FROM AUTHOR]

This article delves into the study of the perturbed cubic-quartic Kudryashov's equation having the Kudryashov's law of refractive index which is recently introduced to the literature. Utilizing the extended rational sine/cosine and sinh/cosh methods, optical solutions such as dark, periodic, and singular are successfully extracted. The two- and three-dimensional plots of these solutions are demonstrated. Besides, the behavior of solitons and the effects of parameters on wave properties are vividly illustrated to explore how the model's parameters influence these solutions. The findings herein can serve as a positive contribution to future research in nonlinear optics, while the methodology employed can be leveraged for similar nonlinear models in real life. [ABSTRACT FROM AUTHOR]

This study investigates the nonlinear Klein–Gordon equation (KGE). We successfully construct some new topological kink-type, non-topological, singular solitons, periodic waves and singular periodic wave solutions to this nonlinear model by using the extended ShGEEM, rational sine-cosine extended (ERSC), and sinh-cosh (ERSCh) methods. In addition, a numerical method for solving the KGE is described in this paper. We use a combination of two numerical techniques called fictitious time integration method and the group preserving scheme (GPS). Fictitious time integration method converts the main equation into a new problem then the GPS is used to gain the numerical solutions. Few experiments are provided to successfully demonstrate the correctness of the approach. [ABSTRACT FROM AUTHOR]

In this article, we presented and investigated a model of the (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation with parabolic law nonlinearity, for the first time. This study intends on two main elements. The first is to construct the parabolic law nonlinearity form of the HFSC model, and the second is to peruse the effect of this form on the soliton wave behavior. Two different analytical techniques, which are a subversion of the new extended auxiliary equation method and the improved generalized Kudryashov method (iGKM), are applicated to construct the soliton solutions of investigated HFSC form. With the help of these methods, bright, singular, and kink soliton forms have been obtained and their characteristics have been depicted by 2D and 3D diagrams. In accordance with our litterateur investigation, the subject of the study and the results obtained have not been reported before and we can say that the results we have attained will be a guide for researchers who want to study in this field. [ABSTRACT FROM AUTHOR]

Purpose: This study includes the examination of the cases where the (1 + 1) -dimensional chiral nonlinear Schrödinger equation also has Bohm potential. This review is not to obtain different soliton solutions for both cases but to obtain a certain type of soliton and to observe the effect of the problem parameters. By using the modified new Kudryashovs scheme. This observation also includes how the soliton behavior is effective by comparing the (1 + 1) -dimensional chiral nonlinear Schrödinger equation (C-NLSE) and (1 + 1) -dimensional chiral nonlinear Schrödinger equation with Bohm potential (C-NLSE-BP), especially by examining the Bohm potential parameter. Methodology: In order to apply the proposed analytical method which is the modified Kudryasovs scheme (m-NKM), as in many studies, the nonlinear partial ordinary differential equation (NLPDE) is first converted into nonlinear ordinary differential equation form (NLODE) by using wave transform. Then, in order to determine which degree the solution function to be proposed will be, the balancing constant is calculated. The next step is to determine the unknown parameters of the problem by applying the m-NKM on NODE, obtaining solution sets, and combine the solution of the Riccati equation, which is the basis of the method, with the proposed solution function and wave transform. Obtaining the optical solution by providing the main NLPDE is the next step that follows this stage. The final stage is the graphical analysis and interpretation of the parameter effect for both problems using the obtained solution function. Findings: The examination of the case with and without Bohm potential was carried out on the behavior of the bright soliton shape, which is one of the basic soliton shapes that many effects can be clearly studied on it. It was successfully shown that both equation parameters and the Bohm potential had a significant effect on the soliton behavior, graphical presentations were made and interpreted in detail. Originality: Such an examination has not been studied before in the literature for the investigated equations. [ABSTRACT FROM AUTHOR]

This article investigates the optical soliton solutions of the Kundu–Mukherjee–Naskar (KMN) equation with the help of the enhanced modified extended tanh method (eMETEM). The equation is crucial for simulating the bending of light beam, rogue waves in the seas, and fiber pulses in optics. Some different kinds of solitons such as singular-periodic, dark and singular for the KMN equation have been successfully obtained by the used method. Utilizing Maple, we confirm that all acquired solutions satisfy the considered equation. We have also analyzed the impact of the parameters graphically in the considered equation. The results shown in the figures reveal that the parameters of the considered equation have an important effect upon soliton dynamics. It is anticipated that the gained solutions and used approach would create remarkable chances for future work in a variety of science areas. [ABSTRACT FROM AUTHOR]

We have extracted some soliton solutions of the fractional longitudinal wave equation with the M-truncated derivative (M-LWE), which emerges in a magneto electro-elastic circular rod. To obtain new results of this model, the unified Riccati equation expansion and new Kudryashov methods have been utilized for the first time. The presented methods have been productively implemented to the considered model. With the help of these two methods, new soliton waves of the M-LWE have been obtained successfully. For a better understanding of the subject and analysis of the results, 3D, contour, and 2D graphs of some soliton solutions have been presented. The interesting part of our work is that both methods, named unified Riccati equation expansion and the new Kudryashov methods have successfully been applied for the first time to get new soliton solutions of M-LWE. [ABSTRACT FROM AUTHOR]

Purpose: In this article, two main subjects are discussed. First, the nonlinear Schrödinger equation (NLSE) with an anti-cubic (AC) nonlinearity equation is examined, which has a great working area, importance and popularity among the study areas of soliton behavior in optical fibers, by using the enhanced modified extended tanh expansion method and a wide range of optical soliton solutions is obtained. Second, the effects of AC parameters on soliton behavior are examined for each obtained soliton type. Methodology: In order to apply the method, the non-linear ordinary differential equation form (NLODE) of the investigated NLSE-AC is obtained by applying the defined wave transformation. Then, with the help of the proposed algorithm for the NLODE form, polynomial form, an algebraic equation system is obtained by setting the coefficients of this form to zero, and the solution of this system is also obtained. After determining the suitable solution set, the optical soliton solution of the investigated problem is obtained with the help of the serial form of the proposed method, a Riccati solution and wave transform. After checking that the solution satisfies the investigated problem, 3D and 2D graphics are obtained for the special parameter values and the necessary comments are made in the relevant sections. Findings: With the proposed method, many optical soliton solutions, such as topological, anti-peaked, combined peaked-bright, combined anti-peaked dark, singular, combined singular-anti peaked, periodic singular, composite kink anti-peaked, kink, periodic and periodic, with different amplitudes are obtained, and 3D and 2D representations have been made. Then, the effect of AC parameters on the soliton behavior in each case has been successfully studied. It has been shown that AC parameters have a significant effect on the soliton behavior, and this effect changes depending on the soliton shape and the parameters. Moreover, providing and maintaining the delicate balance between the soliton shape and the parameters and the interaction of the parameters with each other involves great difficulties. Originality: Although some soliton types of the NLSE-AC equation have been presented for the first time in this study, there is no study in the literature showing the effect of AC parameters on soliton behavior, especially for the NLSE-AC equation. [ABSTRACT FROM AUTHOR]

In this paper, we aim to derive new soliton solutions of (1+1)- and (2+1)-dimensional generalized Sasa–Satsuma equations via the new Kudryashov method. In optical fiber transmission systems, the Sasa–Satsuma equation describes the effects of third-order dispersion, self-steepening and stimulated Raman scattering in the propagation of ultrafast pulses. The considered equations are encountered in various physical applications such as ultra-short and femto-second pulse propagation in optical fibers and dynamics of deep water waves. So, investigation of the novel solutions of the equations is one of the important topics. We have successfully extracted some soliton solutions for the considered equation. The various graphs of the obtained solutions have been depicted in the figures by selecting appropriate parameters. The singular and bright soliton solutions have been revealed in the figures. All acquired solutions have been confirmed to satisfy the considered equations. The results show that the approach may be used to find exact solutions to various nonlinear evolution equations. The new solutions and the paper results may enrich the understanding of the wave propagation in the optical fibers and may shed light on new studies. [ABSTRACT FROM AUTHOR]

This article investigates the optical soliton solutions of the time-fractional Biswas–Arshed (BA) equation, which is a new model for soliton transmission through optical fibers where the eigen-phase modulation is negligible and thus removed. We have studied both models including M-truncated or beta derivative operators. Firstly, the time-fractional BA equations have been transformed into a nonlinear ordinary differential equation using appropriate wave transformations. The singular, singular-periodic, and dark soliton solutions have been derived using the extended rational sine–cosine and sinh–cosh techniques have been applied and the existing conditions for the obtained solutions have been introduced. Utilizing a computer algebraic system, we have verified that all of the derived solutions satisfy the fractional Biswas–Arshed equation. Additionally, we have compared the results for the beta and M-truncated derivatives and have examined how the equation's parameters affect the amplitude of the solitons in 2D and 3D graphical demonstrations. The attained solutions may aid in the comprehension of wave propagation in optical fibers and may contribute to the telecommunication sector. [ABSTRACT FROM AUTHOR]

We examined the (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which arises not only in fluid dynamics, superfluids, physics, and plasma physics but also in the construction of connections between the hydrodynamic and optical model fields. Moreover, unlike the Kadomtsev–Petviashvili equation (KPE), the KP-B equation allows the modeling of waves traveling in both directions and does not require the zero-mass assumption, which is necessary for many scientific applications. Considering these properties enables researchers to obtain more precise results in many physics and engineering applications, especially in research on the dynamics of water waves. We used the modified extended tanh function method (METFM) and Kudryashov's method, which are easily applicable, do not require further mathematical manipulations, and give effective results to investigate the physical properties of the KP-B equation and its soliton solutions. As the output of the work, we obtained some new singular soliton solutions to the governed equation and simulated them with 3D and 2D graphs for the reader to understand clearly. These results and graphs describe the single and singular soliton properties of the (3+1)-dimensional KP-B equation that have not been studied and presented in the literature before, and the methods can also help in obtaining the solution to the evolution equations and understanding wave propagation in water wave dynamics. [ABSTRACT FROM AUTHOR]

The unstable nonlinear Schrödinger equations (UNLSEs) are universal equations of the class of nonlinear integrable systems, which reveal the temporal changing of disruption in slightly stable and unstable media. In current paper, an improved auxiliary equation technique is proposed to obtain the wave results of UNLSE and modified UNLSE. Numerous varieties of results are generated in the mode of some special Jacobi elliptic functions and trigonometric and hyperbolic functions, many of which are distinctive and have significant applications such as pulse propagation in optical fibers. The exact soliton solutions also give information on the soliton interaction in unstable media. Furthermore, with the assistance of the suitable parameter values, various kinds of structures such as bright-dark, multi-wave structures, breather and kink-type solitons, and several periodic solitary waves are depicted that aid in the understanding of the physical interpretation of unstable nonlinear models. The various constructed solutions demonstrate the effectiveness of the suggested approach, which proves that the current technique may be applied to other nonlinear physical problems encountered in mathematical physics. [ABSTRACT FROM AUTHOR]

In this scientific work, we explore the solutions of an important mathematical model appearing in the nonlinear elastic inhomogeneous Murnaghan's rod (NEIMR) or called doubly dispersive equation (DDE) with a useful and powerful analytical technique; which is the extended Kudryashov method with the Bernoulli–Riccati approach. In our research, we have successfully obtained some important solutions of the above-mentioned model such as topological, nontopological, singular solitons, singular periodic and rational solutions. Using the appropriate values of the parameters, we clearly plotted the solutions of the model, 3D, 2D and contour plots. We think that the method used in the study and the results obtained will be an effective motivation tool for many researchers doing research in this field. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated the perturbed Chen–Lee–Liu equation which describes the pulse propagation in the optical fibers, under the impact of the inter-modal dispersion, self-steepening and nonlinear dispersion terms. By using the enhanced modified extended tanh expansion method, bright, singular, periodic singular and periodic bright solitons have been obtained and the effects of the coefficients of the inter-modal dispersion, self-steepening and nonlinear dispersion terms on the soliton's dynamics have been examined in each case. In this respect, the review in the article has not been studied and reported before. The computations throughout this paper have been fulfilled by Maple and also the graphical simulations are demonstrated via Matlab. [ABSTRACT FROM AUTHOR]

In this paper, we have successfully extracted many analytic solutions for the (1+2)-dimensional Chiral non-linear Schrödinger equation (NLSE) by the enhanced modified extended tanh expansion method (eMETEM). The considered method is a recently enhanced version of the classical modified extended tanh expansion method. So, we have successfully extracted the abundant solutions of the (1+2)-dimensional Chiral NLSE. Using a computer algebra system program, we have verified that all derived solutions satisfy the Chiral NLSE. The plots of some solutions are demonstrated to explain the dynamics of the solutions. It is expected that the results of the paper might be helpful for future works in traveling wave theory. [ABSTRACT FROM AUTHOR]

Some new optical solitons find for the Kudryashov equation (KE) in this study. These solutions are in the form of dark, bright, singular, singular-dark solitons and other solutions with certain conditions. These new solutions may be applied in the demonstration of Kudryashov equation in some better way. Modified integration method, the extended Sinh-Gordon equation expansion method (ShGEEM) is utilized to secure the aforesaid solutions. The results are also verified with the use of Mathematica symbolic programming software. These optical solitons suggest that this method is effective and straight forward compared to other methods and the results are reliable. The obtained solutions are very helpful in the field of optical fibers. [ABSTRACT FROM AUTHOR]

In this research article, the Sardar subequation method is used to retrieve new analytical solutions to the space-time local derivative Sasa–Satsuma equation with Atangana's conformable derivative, which defines short pulse propagation in an optical fiber area. This equation is the integrable extension of the nonlinear Schrödinger equation. First, the equation is transformed into an ordinary differential equation utilizing traveling wave transformation. Then, novel different type soliton solutions are acquired using the Sardar subequation approach. The produced soliton solutions play an essential role for scientists in interpreting the physical phenomenon of this equation. Finally, the graphs of some solutions are depicted at appropriate values of parameters. The achieved results show the simplicity, reliability, and potentiality of the proposed method. [ABSTRACT FROM AUTHOR]

In this study, two integration techniques are employed to reach optical solitons to the (n + 1) -dimensional nonlinear Schrödinger's equation ((n + 1) -NLSE) with Kerr and power laws nonlinearities. These are the undetermined coefficient and Bernoulli sub-ODE methods. We acquired bright, dark, and periodic singular soliton solutions. The necessary conditions for the existence of these solitons are presented. [ABSTRACT FROM AUTHOR]

The aim of this study is to examine the effects of the coefficients of group velocity dispersion, self-frequency shift and self-steepening terms on the optical solitons of the (3+1)-dimensional Sasa–Satsuma equation which is important for modeling many physical events and it has recently been introduced to the literature. To get desired results we have used the solution approach as following. The first step is defining the wave transform and applying it to the nonlinear partial differential equation (NLPDE). Then we have obtained the ordinary differential equation (ODE). Afterwards it is introduced and applied the new Kudryashov method (nKM) and unified Riccati equation expansion method (UREEM) to obtain the algebraic equation system. Finally, by determining the parameters from the solution of algebraic system we have substituted them in the solution function. By using the solution function we have obtained interesting soliton graphics under different parameter values. In addition to the basic solitons of the (3+1)-dimensional Sasa–Satsuma equation (SSE), V-shaped and periodic singular type solitons have also been obtained. The findings of the soliton behaviors under the parameter values effect has not been studied before. The results presented in the article are reported for the first time. • The aim of this study is to examine the effects of the eigenfrequency shift and self-perpendicularity coefficients on the optical solitons of the (3+1) dimensional Sasa–Satsuma equation. • New Kudryashov (nKM) and unified Riccati equation expansion methods have been used. • The most important output of the study is that detailed investigations were made on the effects of the coefficients of group velocity dispersion, self-frequency shift, and self-steepening terms on the soliton behavior, showing that they have a significant impact. • This study will help future research in the optic field of physics. [ABSTRACT FROM AUTHOR]

In this paper, optical solitons of higher-order nonlinear Schrödinger equation with Kudryashov's sextic power-law of nonlinear refractive index are investigated via the direct mapping method. The considered model identifies the optical soliton pulse propagation in the optical fibers. Deriving the optical solutions of investigated model such as sextic power is critical but difficult work. The primary aim of this paper is to graphically examine the impact of power-law nonlinearity (pLawNL) and chromatic dispersion (CD) parameters clarifying self-phase modulation (SPM) in the equation on soliton behavior as well as obtaining optical soliton solutions. To according to the used technique, we first used the complex wave transform to generate the nonlinear ordinary differential equation (NLODE) form of the nonlinear Schrödinger equation (NLSE) with Kudryashov's sextic power-law the nonlinear refractive index (SPLawNRI). Then, we were able to produce a system of linear equations in polynomial form by using the approach. Different solution sets including the values of the parameters of the studied equation and the suggested approach were produced by solving the linear system of equations. We acquired the optical soliton solutions of the main equation after inserting the sets and wave transformation into the solution functions suggested by the approach. The constraint conditions for the related solutions were suggested. We proved that the gained solutions satisfied the NLSE with Kudryashov's SPLawNRI under the suggested constraint conditions. We present contour, 3D and 2D depictions in various simulations in figures to comment the obtained solution functions. Besides, we investigate the effects of the power-law nonlinearity parameter that expresses SPM in the main equation and the parameter that are the group velocity dispersion (GVD) or chromatic dispersion (CD) on soliton behavior. The results suggest that the utilized approach is efficient, reliable, and powerful to be readily applied to various NLSEs with higher-order or higher pLawNLs that characterize real-life problems. [ABSTRACT FROM AUTHOR]

This manuscript addresses to investigate the optical soliton solution of the Manakov model which governs soliton transmission technology by utilizing the auxiliary equation technique. After obtaining the nonlinear ordinary differential equation form (NODE) of investigated nonlinear partial differential equation problem (NLPDE) with the complex wave transform (cWT), the serial form is suggested in accordance with the utilized auxiliary method, and after written in the NODE form, the linear algebraic system is formed and by aid of the any symbolic calculation package the unknown parameters are obtained. In order to gain the soliton solutions, the proposed series form, the solution functions of the method, and the cWT are brought together and it is checked that the obtained solutions satisfy the main equation. By considering the functions that satisfy the main equation as solutions, graphical presentations are made and interpreted in order to better understand the obtained results. By applying the effective auxiliary equation method, optical soliton pulse solutions such as bright, singular, V-shaped, periodic singular, periodic bright, periodic bright–dark have been obtained, and it has been shown that the method can be applied effectively for such equations. The auxiliary equation method has not been applied to the Manakov model and the V-shaped, periodic singular, periodic bright soliton pulse solutions have not been reported before. [ABSTRACT FROM AUTHOR]

This article aims to introduce the generalized stochastic perturbed Schrödinger–Hirota equation, which incorporates multiplicative white noise in the Itô sense. The study focuses on investigating the stochastic optical soliton solutions related to the governing model to understand the characteristics and properties of these soliton solutions. We can achieve this by utilizing the enhanced Kudryashov method and the enhanced direct algebraic approach. The study stands out for its innovative approach to examining the effects of multiplicative white noise on stochastic optical soliton solutions through the lens of the generalized stochastic perturbed Schrödinger–Hirota equation. Utilizing the enhanced Kudryashov method alongside the novel enhanced direct algebraic method, this research effectively identifies and delineates a range of soliton solution types, such as bright, dark, singular, and straddled solitons. This thorough investigation not only underscores the efficacy and efficiency of these methods but also sheds light on the impact of noise on the structural and dynamic properties of soliton solutions, supported by illustrative visualizations. A notable finding is that all solutions derived using the enhanced direct algebraic method feature a nonzero coefficient for the highest-order term of the auxiliary equation. The diversity of solutions obtained through these approaches includes bright, dark, singular, and straddled solitons. Additionally, the study reveals Weierstrass and Jacobi elliptic doubly periodic solutions, which can transition into various soliton types under specific parameter conditions, offering a broader understanding of soliton behavior and properties. [ABSTRACT FROM AUTHOR]

This article presents an investigation of soliton solutions for the complex cubic–quintic Ginzburg–Landau equation having anti-cubic law nonlinearity which arises in wave propagation through optical fibers using the enhanced Kudryashov method for the first time. The used method generates all the solutions extracted by classical Kudryashov and new Kudryashov methods at once. This simple yet powerful technique can be applied to a broad range of nonlinear partial differential equations. Applying the method to the Ginzburg–Landau equation, dark, bright, and kink solitons were successfully obtained. Moreover, the effect of the anti-cubic law form, which is the nonlinearity form of the model, on the dynamics of the soliton was examined and it was observed that this form caused a deterioration from a dark soliton to a W-like soliton. Both types of solitons are important in nonlinear systems and are monitored in a variety of physical systems such as optical fibers, Bose–Einstein condensates, and superconductors. This study may have valuable insights into the behavior of soliton solutions of the complex cubic–quintic Ginzburg–Landau equation having anti-cubic law nonlinearity and can contribute to the development of new technologies in nonlinear optics. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate analytic soliton solutions of a nonlinear Schrödinger equation (NLSE), including Kudryashov's sextic power-law nonlinearity by introducing different approaches of two efficient analytical methods. The considered equation has been recently introduced by N. A. Kudryashov to describe pulse propagation in optical fibers. The refractive index in the equation comprises six terms, each of the terms containing a power-law component. Applying a wave transformation to the considered NLSE and splitting up the real and imaginary parts, the NLSE is converted to the nonlinear ordinary differential equations (NLODEs). Then, the solutions of NLODEs are considered as suggested in the proposed method and suggested solutions that include some unknown parameters are substituted into the NLODE. An algebraic equations system is acquired by collecting the same power of the unknown function and equating all coefficients to zero. The unknown parameters in the system, and so the solutions of the NLSE, are found by solving the system. In the proposed first method, the modified extended tanh method is enhanced by proposing more solutions. The proposed second method, a different version of auxiliary methods, can remarkably reduce calculations to easily get solutions for the NLSEs with higher-order or power-law nonlinearity. The two proposed methods are successfully applied to the considered NLSE and the abundant solutions of the NLSE are attained. Besides, 2D, 3D and contour graphs are demonstrated in figures for the physical illustrations of the gained solutions. Obtaining the solutions of NLSEs with higher-order or power-law nonlinearity has crucial importance but still challenging work. So, we propose different approaches of two efficient analytical methods, namely, enhanced modified extended tanh expansion method and an auxiliary function method. The derived results imply that the used methods are very efficient, reliable and powerful such that they can be easily implemented to many nonlinear NLSEs with higher-orders or higher power-law nonlinearities that describe real-life phenomena. [ABSTRACT FROM AUTHOR]

In this paper, we study the novel exact solutions of the (2 + 1) -dimensional Biswas–Milovic equation for the description of pulse propagation in optical fiber. We successfully constructed some important solutions, such as dark, singular, combined dark-singular soliton, singular periodic wave and rational function solutions, have been analytically obtained using the extended tanh method. These solutions might play an important role in the engineering and physics fields. It is shown that the considered method provide a straightforward and powerful mathematical tool for solving problems in nonlinear optic and other physical phenomena. [ABSTRACT FROM AUTHOR]