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This study retrieves some novel exact solutions to the family of 3D space–time fractional Wazwaz–Benjamin–Bona–Mahony (WBBM) equations in the context of diverse nonlinear physical phenomena resulting from water wave mechanics. The family of WBBM equations is transformed for this purpose using a space and time fractional transformation into an ordinary differential equation (ODE). The ODE then uses a strong method, namely the Unified Method. Consequently, lump solutions, dark-bright soliton, singular and multiple soliton solutions, and periodic solutions are investigated. The disparities between the current study's conclusions and previously acquired solutions via other approaches are examined. All wave solutions produced are determined to be novel in terms of fractionality, unrestricted parameters, and implemented technique sense. The impact of unrestricted parameters and fractionality on the obtained solutions are visually presented, along with physical explanations. It is observed that the wave portents are varied with the increase of unrestricted parameters as well as fractionality. We dynamically showed that the appropriate transformation and the applied Unified approach more proficient in the study of water wave dynamics and might be used in future researches to clarify the many physical phenomena. The novelty of this work validate that the proposed method seem simple and useful tools for obtaining the solutions in PDEs and it is expected to use in mathematical physics and optical engineering. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is one of the most important physical model in optical fiber theory for comprehension of the fluctuations of optical bullet development. In this study, the exact bullet solutions for the (3+1)-dimensional Schrödinger equation which demonstrate the bullet behaviours in optical fibers can be accumulated through the Sardar sub-equation method and the unified method. The applied strategies may retrieve several kinds of optical bullet solutions within one frameworks as well as is quite simple and reliable. Mathematica are utilised for describing the dynamics of different wave structures as 3D, 2D, and contour visualisations for a given set of parameters. As a result, we are able to develop a variety of travelling wave structures namely the periodic, singular and V shaped soliton wave solutions. The stability analysis for the derived results is analysed efficiently while the modulation instability for the governing model has also been studied to demonstrate the reliability of the research. The approaches implemented here works perfectly and can be extended to deal with many advanced models in contemporary areas of science and engineering. The solutions attain by using these techniques are robust, unique and straight forward and has applications in different fields of physics, engineering and mathematical science. Specially physical applications of these obtain results are in the transmission of data in optical fibers. We also add the graphics for the better understanding of the attain solutions behaviour. [ABSTRACT FROM AUTHOR]

Abstract This study retrieves some novel exact solutions to the family of 3D space–time fractional Wazwaz–Benjamin–Bona–Mahony (WBBM) equations in the context of diverse nonlinear physical phenomena resulting from water wave mechanics. The family of WBBM equations is transformed for this purpose using a space and time fractional transformation into an ordinary differential equation (ODE). The ODE then uses a strong method, namely the Unified Method. Consequently, lump solutions, dark-bright soliton, singular and multiple soliton solutions, and periodic solutions are investigated. The disparities between the current study's conclusions and previously acquired solutions via other approaches are examined. All wave solutions produced are determined to be novel in terms of fractionality, unrestricted parameters, and implemented technique sense. The impact of unrestricted parameters and fractionality on the obtained solutions are visually presented, along with physical explanations. It is observed that the wave portents are varied with the increase of unrestricted parameters as well as fractionality. We dynamically showed that the appropriate transformation and the applied Unified approach more proficient in the study of water wave dynamics and might be used in future researches to clarify the many physical phenomena. The novelty of this work validate that the proposed method seem simple and useful tools for obtaining the solutions in PDEs and it is expected to use in mathematical physics and optical engineering.

In this research article, the dynamics of optical solitons in cascaded system with spatio-temporal dispersion is discussed. The Kerr medium is considered for this investigation. Two versatile integration tools have been used to extricate different traveling wave solutions. Rapid convergent approximation method has been employed to extract bright, singular and periodic solitons whereas polynomial function solutions are obtained by unified method. The constraints on the parameters for the validity of these solutions are also provided. Galilean transformation is prosecuted to construct the planer dynamical system of the said model. Runge–Kutta fourth-order technique is used to extract the non-linear periodic waves of the considered problem and consequences are depicted through different portraits. Further, by employing an extrinsic periodic force, chaotic and quasiperiodic pattern of discussed model is analyzed for different values of the parameters. While chaotic and quasiperiodic behaviors are observed for some values of parameters of considered system by keeping the force and frequency of the perturbed dynamical system fixed. Sensitive anatomy is carried out for diverse initial value problems. It is investigated that the chaotic and quasiperiodic pattern is present in the perturbed dynamical system. [ABSTRACT FROM AUTHOR]

In this work, the new nonlinear (1 + 1) -dimensional Schrödinger (NLS) equation, which describes wave propagation in optical fibers, has been successfully studied using the (G ′ / G) -expansion method and the Unified method (UM). These techniques have been applied for the first time in extracting new analytic solutions for the mentioned NLS equation. The obtained solutions exhibit various types of optical solitons, including singular solitons, solitary waves, periodic solitons, and wave collisions. These solutions have wide-ranging applications in various fields such as fiber optics, telecommunication systems, plasma physics, hydrodynamics, and nonlinear optics. The behavior of the derived solutions has been visually represented using three-dimensional, two-dimensional, and contour plots, considering appropriate choices of the involved parameters within specific time intervals. These plots provide insights into the dynamics and characteristics of the solutions. The effectiveness, reliability, and ease of application of the (G ′ / G) -expansion method and the UM are demonstrated through the obtained solutions. These mathematical tools have proven to be powerful in solving nonlinear partial differential equations, offering valuable contributions to the field of mathematical physics, soliton theory and nonlinear sciences. [ABSTRACT FROM AUTHOR]

In this manuscript, we employ the unified method and the Sardar subequation method to systematically analyze various wave structures within the (3+1)-dimensional extended quantum nonlinear Zakharov–Kuznetsov equation, incorporating test function approaches. The equation, integral to understanding the intricate dynamics of quantum plasma in diverse scenarios like astrophysical environments, fusion devices, space plasma, and quantum fluids, serves as a foundation model for studying nonlinear waves, structures, and the collective quantum behavior of particles. Our study yields a spectrum of solutions, including bright, dark, singular, periodic, kink, and anti-kink solutions, and we conduct a modulation instability analysis through linear stability techniques. Furthermore, we elucidate the physical implications of our findings using 3-dimensional, 2-dimensional, and contour profiles with relevant parameters, exclusively implementing the symbolic software Mathematica for computations. The obtained results exhibit exceptional merit when compared to existing literature, showcasing the effectiveness of our approach in providing a robust and intensive mathematical framework for addressing complex nonlinear wave phenomena. [ABSTRACT FROM AUTHOR]

The problem studied in the quadratic-cubic fractional nonlinear Schrödinger model in superfast fibers is the propagation of optical pulses in fiber optic systems. Optical pulses in fiber optic systems are important because they enable the transmission of information over long distances at high speeds with minimal loss and interference. We utilized the unified method and the extended simple equation method. These methods have been instrumental in obtaining important results in the study of the quadratic-cubic fractional nonlinear Schrödinger model in superfast fibers. These methods have allowed researchers to gain a deeper understanding of the dynamics of optical pulses and have contributed to advancements in fiber optic communication systems. Superfast fibers have revolutionized data transmission by providing high-speed and reliable connectivity. They enable faster internet connections, support emerging technologies, and have implications for various industries. While there are challenges associated with their deployment and maintenance, the benefits of superfast fibers outweigh these limitations. One of the key novelties of the this model is its ability to accurately describe the propagation of ultrafast optical pulses in highly nonlinear fibers. The model takes into account both quadratic and cubic nonlinear effects, which are known to play crucial roles in the dynamics of intense light pulses. [ABSTRACT FROM AUTHOR]

This paper investigates different kinds of exact solutions of variable-coefficients Chiral Schrödinger equation (VCCNLSE). In this equation, the fractional quantum Hall effect edge states are described. By the unified method, we successfully get the soliton, solitary and elliptic wave solutions. Through the (G ′ / G) -expansion and the exponential expansion methods, we acquire different traveling wave solutions. Using the extended F -expansion method, we attain the Jacobi elliptic function solutions. So as to get more physical information about the exact solutions of VCCNLSE, 3D, 2D , density and contour maps are described. [ABSTRACT FROM AUTHOR]

This research delves into a comprehensive exploration of exact solutions for the time-space fractional phi-four equation through the implementation of both the unified method and the modified Khater method. The Phi-four equation, rooted in the Klein-Gordon model, holds significance in diverse scientific realms, such as biological and nuclear physics, making it imperative to unravel its solutions. Leveraging a well-suited wave transformation, the governing equation is transformed into an ordinary differential equation, laying the groundwork for an in-depth analysis. The proposed methodologies yield a rich spectrum of solutions, encompassing trigonometric, hyperbolic, and rational forms. These solutions manifest as Kink, bright singular, dark, combo dark bright, periodic and singular periodic solutions, each characterized by explicit conditions defining their validity. The versatility and computational robustness of these methods establish a robust framework for soliton research. To provide a comprehensive assessment, a comparative analysis is conducted, contrasting the outcomes obtained through the proposed methodologies with those derived from earlier approaches. A key fact of this study is the visual representation of a subset of these solutions through carefully crafted graphical illustrations. Parameter values are judiciously selected to elucidate the nuanced characteristics of the solutions, enhancing our understanding of the model's physical behavior. It is noteworthy that all computations are meticulously executed with Mathematica software, underscoring the precision and computational efficiency integral to this research. The novelty of this paper lies in the identification of new solutions through the application of the unified and modified Khater methods. These solutions are not previously documented in existing literature. The visual depictions of dynamic behaviors through two and three-dimensional figures further contribute to the originality of our findings. [ABSTRACT FROM AUTHOR]

For the sake of intellectual curiosity, in this manuscript we analyse the soliton solutions of a dynamical model namely, the (2 + 1) -dimensional generalised Bogoyavlensky–Konopelchenko equation arising in mathematical physics. We apply the Hirota bilinear method and the unified method to produce a variety of novel solutions. By using the aforementioned techniques, we establish fresh sorts of solutions for the concern problem, including breather wave, lump-periodic, two wave solutions, dark, periodic, and rational soliton solutions. In addition, we describe the stability analysis of our chosen model. For certain values of the required free parameters, the propagation of the derived well-furnished outcomes is visualised in different profiles using Mathematica 13.0. All of the findings in this study are crucial to comprehending the physical significance and behaviour of the investigated equation, which highlights the significance of examining various nonlinear wave phenomena in mathematical physics. The acquired results demonstrate that the computational strategies used are efficient, competent, and concise and may be used in conjunction with representative computations to apply to more complicated phenomena. [ABSTRACT FROM AUTHOR]

In this study, we solve fractional order PDEs with free parameters that regulate wave motion in a low-pass electrical transmission line equation (LPET) using the unified technique. To convert the governing equation into an ordinary differential equation (ODE), we have used a simple linear fractional transformation. The unified technique can therefore be used to find various single and extracting wave solutions to the equation of attention. Using Maple-18 software, all of the established solutions are validated and verified. By selecting appropriate values for the fractional order and free parameters, the two dimensional (2D) and three dimensional (3D) profiles of some of the assimilation solutions demonstrating kink, singular kink, cross kink, bright-dark kink, lump type kink, anti-kink, dark-anti kink, singular periodic, periodic, singular soliton solutions are displayed. The effect of free parameters and fractionality on the dynamical properties of the precise and exploited solutions are graphically illustrated and discussed in detail. We have also compared our results to those found in the literature review. Wherever possible, the judgment demonstrates no significant differences between our realized and published solutions. The results of this investigation show that the adopting technique is both productive and powerful in extracting wave solutions to nonlinear evolution problems that arise in the areas of applied mathematics, mathematical physics, and engineering.

In the intricate realms of quantum field theory, cosmology, and condensed matter physics, the Phi-4 model emerges as a potent catalyst, shaping mathematical landscapes with profound implications. Our research has uncovered a set of soliton solutions that were previously unreported in the existing literature. This transformative journey has allowed us to unravel a rich tapestry of new insights in the field. Our findings will contribute significantly to advancing knowledge in this area of study. Through innovative analytical approaches, we breathe life into these solitons, delving into their dynamic behaviors and unveiling the nuanced influence of factors such as wave velocity (ω) and diverse free parameters. The elegance of our methodologies lies in their simplicity, offering accessible pathways for exploration. As we stand on the threshold of discovery, the solutions and intricate wave profiles we present broaden our current understanding and beckon further investigation, resonating with the Phi-4 model’s enduring melody that harmonizes across scientific frontiers.

This paper reveals the conformable fractional space-time perturbed Gerdjikov-Ivanov (GI) equation, which is applied to nonlinear fibre optics together with photonic crystal fibres. The foremost intent of our operation is adopting the unified method, the modified F-expansion method and the modified Kudryashov method to hunt for definite solutions to the equation. In addition, by selecting fairish values, fluctuation behaviours of the solutions are drafted.

The study aims to explore obliquely propagating optical wave solutions to the (2 + 1)-dimensional chiral nonlinear Schrödinger (NLS) equation in both the absence and presence of the Atangana derivative. In order to convert the classical-order chiral nonlinear Schrödinger equation to an ordinary differential equation, a transformation associated with wave obliqueness is applied. Hereafter, the unified method is applied to the reduced equation. As outcomes, the dark, periodic singular with unequal wave length, periodic with equal wavelength, periodic, unsmooth periodic, singular periodic soliton solutions are received to the ordinary differential equation. Later, the acquired solutions are then put to the applied transformation associated with obliqueness. Moreover, the fractional-order chiral NLS equation is solved by using the spatiotemporal Atangana derivative with oblique wave transformation and the unified method. In terms of wave obliqueness, fractionality, and applied technique sense, all generated wave solutions are revealed to be novel. Along with their physical explanations, the impacts of obliqueness and fractionality on the solutions are graphically illustrated. It is exposed that as obliqueness and fractionality increase, the optical wave phenomena change. Additionally, it is discovered that the employed method can be used to obtain novel optical soliton features to the chiral nonlinear Schrödinger equation with or without fractional and obliqueness constraints. It can be assured that the utilized method is more powerful than the other methods. As a result, the method can be used in future research to explain the many physical phenomena that arise in optical fiber communication networks.

In this paper, new solutions of the time-fractional Hirota–Satsuma coupled KdV equation model the intercommunication between two long waves that have well-defined dispersion connection received successfully by the unified method, the improved F -expansion method and the homogeneous balance method. In contrast, these methods are simple and efficient, and can obtain different exact solutions to this equation. By symbolic calculation, polynomial solutions, hyperbolic function solutions, trigonometric function solutions, rational function solutions, etc. are acquired. Furthermore, we plot and analyze some solutions. [ABSTRACT FROM AUTHOR]

In this article, we investigate the generalised version of the nonlinear Schrödinger equation namely the fractional Schrödinger–Hirota (NLFSH) equation with third order dispersion and Kerr law of nonlinearity, which describes the dynamics of optical solitons in a dispersive optical fiber. An amelioration of the approaches, namely the improved F-expansion approach and the unified method, are used to formulate the abundant optical solitons. After that, utilizing the aforementioned techniques and computational software, different optical solitons are retrieved, including dark, singular, periodic, rational, hyperbolic solitary wave, and trigonometric function solutions. Secondly, we discuss the stability analysis of our selected model which confirm that the governing model is stable. Additionally, the acquired results demonstrate that the suggested strategies have a significant ability to successfully acquire numerous fresh soliton type solutions for the NLFSH equation. For certain values of the required free parameters, the dynamical behaviours of these solutions are visualised in 2D and 3D using Mathematica 13.0. The acquired findings demonstrate the power, effectiveness, and simplicity of the suggested strategies for finding novel solutions to diverse classes of nonlinear partial differential equations in optical engineering and applied sciences. [ABSTRACT FROM AUTHOR]

The Tzitzéica equation, the Dodd–Bullough–Mikhailov equation, and the Tzitzéica–Dodd–Bullough equation arise in various branches of science and technology, such as solid state physics, nonlinear dynamics, nonlinear optics, and quantum field theory problems. In this paper, we discuss how to find more general wave solutions of these three nonlinear evolution equations with physical applications using the unified method. The unified method straightforwardly gives many more general solutions with free parameters without the need for extra hardware support. After performing calculations, the graphs of the solutions are plotted using Maple to give insight into the physical structure of the wave solution. Only some selected solutions are plotted to visualize their behavior. Considering the importance of having solutions for nonlinear wave phenomena, we clearly see that the method has many merits and demonstrates a comprehensive application to obtaining solutions in an efficient way. In addition, diverse types of geometrically structured solitons such as an anti-bell-shaped soliton, a flat soliton, a kink, and a singular soliton are produced by using arbitrary parameters. [ABSTRACT FROM AUTHOR]

The purpose of this study is to introduce a new extension method named the generalized unified method (GUM) and apply this method to the the (2+1) dimensional Kundu–Mukherjee–Naskar (KMN) equation. The GUM as a powerful method provides more general exact solutions for nonlinear partial differential equations (NPDEs) in a compact form with free parameters. Various exact solutions including also hyperbolic, trigonometric and rational forms can be derived from the obtained exact solutions with tuning these free parameters. The reason of choosing the KMN equation is that this equation as extension of the Schrödinger equation is used to model great numbers of considerable physical phenomena such as ion-acoustic waves in oceanic rogue waves, magnetized plasmas, bending of light beams, propagation pulses in optical fiber. Considering the physical importance of the KMN equation, the obtained wide range solution sets by using the GUM will play a significant role in the applied sciences that use the KMN equation to model their problems. [ABSTRACT FROM AUTHOR]

In this study, the unified and improved F-expansion methods are applied to derive exact traveling wave solutions of the simplified modified Camassa-Holm (SMCH) equation. The current methods can calculate all branches of solutions at the same time, even if several solutions are quite near and therefore impossible to identify via numerical methods. All obtained solutions are given by hyperbolic, trigonometric, and rational function solutions which obtained solutions are useful for real-life problems in uid dynamics, optical fibers, plasma physics and so on. The two-dimensional (2D) and three-dimensional (3D) graphs of the obtained solutions are plotted. Finally, we can state that these strategies are extremely successful, dependable, and simple. These ideas might potentially be applied to many nonlinear evolution models in mathematics and physics. [ABSTRACT FROM AUTHOR]

We present a paper on the different methods for finding solutions to the Caudrey–Dodd–Gibbon equation with variable coefficients, which has wide applications in quantum mechanics and nonlinear optics. We have studied the equation analytically by using the unified method and the modified Kudryashov method. With the aid of symbolic computation and several types of auxiliary equations, we have obtained soliton solutions and other solutions. Then, we assign different values to the parameters, generating two-dimensional and three-dimensional graphics of the solutions, and discuss the interaction of several groups of solion solutions to form periodic wave solutions and kink solitary wave solutions.

This paper reveals the conformable fractional space-time perturbed Gerdjikov-Ivanov (GI) equation, which is applied to nonlinear fibre optics together with photonic crystal fibres. The foremost intent of our operation is adopting the unified method, the modified F-expansion method and the modified Kudryashov method to hunt for definite solutions to the equation. In addition, by selecting fairish values, fluctuation behaviours of the solutions are drafted. [ABSTRACT FROM AUTHOR]

This study is used to investigate the exact explicit solutions and dynamical behaviors of the (1+1)-dimensional integrable system of Drinfel’d–Sokolov–Wilson equations in dispersive media. Firstly, the unified method is implemented to find explicit solutions in polynomial and rational forms. These solutions include rational, dark and bright soliton structures. After that, the planar dynamical system of the considered model is obtained using the Galilean transformation. The phase portraits of the bifurcations are drawn from the planar dynamical system for different physical parametric values using the fourth-order Runge–Kutta method. Further, an external force is imposed on the dynamical system to observe the phase portraits of chaotic trajectories. These tracks are outlined for different values of the strength and frequency of the external force acting on the dynamical system. The outcomes show that a given model has chaotic behvior as its solution becomes disordered by taking small variation in the strength and frequency of the external force. The results are new and can be helpful in investigating the dynamical behaviors of the other nonlinear physical models arising in optics, bio-mathematics and so many other fields of science.

This study investigates some analytic solutions and phase portraits to the diffusive predator-prey system in studying the spatiotemporal dynamics of a predator-prey community in ecology through an analytical approach and a qualitative theory of planar dynamical systems, respectively. To accomplish such aims, a simple wave transformation is applied to the diffusive predator-prey system for converting it into a system of ordinary differential equations with a planar dynamical system for analyzing the behavior of bifurcation properties and analytic solutions. Then, the analytical unified method is employed to the attained system. The applied wave transformation is put back to the obtained solutions of the system of ordinary differential equations. Finally, the analytic solutions namely, kink, anti-kink, singular, periodic-singular, and sinusoidal wave solutions are attained to the considered system. All the constructed wave solutions are found to be new from the viewpoint of the application of the unified method. In order to verify the biological wave phenomena of predator and prey populations, some graphs are presented for illustrating the analytic solutions, which have interesting implications in ecology. The effects of free parameters and wave celerity on the attained solutions are demonstrated graphically along with their physical descriptions. The graphical outputs reveal that the predator and prey population densities are changed with the change in the free parameters. Wave solutions to the fractional diffusive predator-prey system are also reported with Atangana conformable derivative sense. As the value of the fractional parameter increases, the smoothness of the anti-kink wave profile is found to increase gradually, but the steepness decreases. For the sinusoidal wave profile, the periodicity and smoothness increase as the value of the fractional parameter increases. Thus, the present study may enrich the interpretation of the spatiotemporal dynamics of predator-prey interaction in a real environment. It is also found that the applied method and the relevant transformation are effective and easy to use for acquiring new analytic solutions to the diffusive predator-prey system over the other analytic methods. The method of interest is novel and efficient because it overcomes the weaknesses and deficiencies of the other analytic methods. Therefore, the method can be applied to further studies to explain various physical phenomena arising in ecology.

The aim of this work is to find some intriguing optical soliton solutions in (2 + 1) dimensions. These soliton solutions including rational, dark, periodic, and elliptic solitons are discovered using the unified technique and the fractional order Local M-derivative to address the temporal fractional Kundu–Mukherjee–Naskar equation. It is the modification of familiar Nonlinear Schrödinger equation and used to describe the bending of an optical solitonic beam in the domain of nonlinear fiber optics and communication system. The obtained solutions are suggested with relevant conditions for their existence and displayed against 3D graphics. Also, to observe and identify the effect of fractional-order parameter on constructed solutions is shown through 2D graphs. The findings highlight that the suggested approach is simple, efficient and successful in determining the exact solution of models in optics, engineering, and other nonlinear sciences. [ABSTRACT FROM AUTHOR]

In this paper, we explore the generalized (3 + 1)-dimensional modified Kadomtsev–Petviashvili equations with variable coefficients, which are usually used in the fields of ferromagnetism, magneto-optics, plasma physics and fluid mechanics. For the sake of uncovering more physical phenomena related to this system, we consider the single-traveling-wave polynomial solutions in the light of the unified method and the double-wave polynomial solutions in line with the generalized unified method. Remarkably, the solitary-, soliton- as well as elliptic-type solutions are all discussed in these two kinds of solutions. Furthermore, the physical explanations of the solutions are given graphically and analytically for different choices of the free parameters (especially of the nonlinear coefficients of the equations that related to the physical insights). By discussing the wave propagation of each solution that we procured from the perspectives of amplitude, shape, symmetry or periodicity, we are capable of realizing the inherent characteristics of this equation commendably and discovering the correlative physical world more efficiently. [ABSTRACT FROM AUTHOR]

This key purpose of this study is to investigate soliton solution of the fifth-order Sawada–Kotera and Caudrey–Dodd–Gibbon equations in the sense of time fractional local M -derivatives. This important goal is achieved by employing the unified method. As a result, a number of dark and rational soliton solutions to the nonlinear model are retrieved. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behavior. The results demonstrate that the presented approach is more effective in solving issues in mathematical physics and other fields. [ABSTRACT FROM AUTHOR]

We present a paper on the different methods for finding solutions to the Caudrey–Dodd–Gibbon equation with variable coefficients, which has wide applications in quantum mechanics and nonlinear optics. We have studied the equation analytically by using the unified method and the modified Kudryashov method. With the aid of symbolic computation and several types of auxiliary equations, we have obtained soliton solutions and other solutions. Then, we assign different values to the parameters, generating two-dimensional and three-dimensional graphics of the solutions, and discuss the interaction of several groups of solion solutions to form periodic wave solutions and kink solitary wave solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the unified method and the modified Kudryashov method for obtaining exact traveling wave solutions of conformable fractional partial differential equations. In addition, a connection is given between the two methods. Then, using these methods, we obtain new exact solutions for the space-time fractional (2 + 1)-dimensional Calogero-Bogoyavlensky-Schiff equation. Fractional derivatives are described in conformable sense. Various solutions have been obtained, including one-soliton, kink, anti-kink, periodic wave solutions, and multiple-soliton solutions. We also provide a graphical representation of some interesting exact solutions to the equation and discuss the behavior of these solutions. The considered methods can be effectively applied to a wide range of nonlinear fractional partial differential equations. [ABSTRACT FROM AUTHOR]

This present study examines the closed-form wave results (CFWRs) of the nonlinear time-fractional model (NLTFM) with Atangana’s conformable fractional derivative (ACFD) through the unified process. ACFD and its properties are included. Then, by proposing the studied approach, the CFWRs of NLTFMs, displayed with the ACFD, are classified. This model defines the transmission of optical solitons in physics and applied mathematics and may be acute to the soliton waves. A lot of new CFWRs have been gained. For a few parameter values involved in the answer functions, physical behaviors on 3D graphics, contour graphics, and 2D graphics for the different values of αwere examined. We apply the unified process on the NLTFM model and provided thirty six CFWRs. Contrary, Y. Gurefe [40] studied the same model model to derive only five CFWRs. As results, our technique in the present paper provides abundant CFWRs. From the observation of the outcomes acquired, it is noticed that the studied process can generate essential effects in taking the closed-form wave results of NLTFMs containing ACFD.

The primary purpose of this study is to analyze non-autonomous femtosecond waves with various geometrical configurations correlated to the generalized derivative nonlinear Shrödinger equation (NLSE) with variable coefficients. Numerous academic publications, especially in nonlinear optics, material science, semiconductor, chemical engineering, and many other fields, have looked into this model since it is closer to real-world situations and has more complex wave structures than models with constant coefficients. It can serve as a reflection for the slowly altering inhomogeneities, non-uniformities, and forces acting on boundaries. New complex wave solutions in two different categories are proposed: implicit and elliptic (or periodic or hyperbolic) forms are obtained for this model via the unified method. Indeed, the innovative wave solutions that were achieved and reported here are helpful for investigating optical communication applications as well as the transmission characteristics of light pulses.

This paper is devoted to addressings the fairly interesting soliton solutions for the time fractional combined Korteweg-de Vries-modified Korteweg-de Vries equation (KdV–mKdV equation) and modified Burgers-KdV equation. The unified method along with conformable, Beta and local M-derivative are used to construct the general structure of solitary wave soliton solutions. The method allows us to find solutions in both polynomial and rational forms. Further, the comparison of solutions are given out through 3D and 2D-plots to expose the impact of fractional parameter on the obtained solutions. The reported solutions are novel and have not been discussed in the literature.

In this work, we consider the (3+1) dimensional conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Solitary wave solutions, soliton wave solutions, elliptic wave solutions, and periodic (hyperbolic) wave rational solutions are obtained by means of the unified method. The solutions showed that this method provides us with a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences.

This work studies two important temporal fractional nonlinear evolution equations, namely the (2+1)-dimensional Chaffee–Infante equation and (1+1)-dimensional Zakharov equation by way of the unified method along with properties of local M-derivative. The typical structures of fractional optical soliton wave solutions are obtained in polynomial and rational forms. Further, to grant the validity of non-singular solutions are given with limitation conditions and graphically depicted in 3D. Also, to expose the effect of a local fractional parameter on expected non-singular solutions are depicted through 2D graphs. The predicted solutions are revealing that the proposed approach is straightforward and valuable to find the solitary wave solutions of other nonlinear evolution equations.

In this article, the unified method, the improved F-expansion method and the modified Kudryashov method are used to obtain the traveling wave solutions of the generalized Zakharov–Kuznetsov equation with variable coefficients. Different types of traveling wave solutions are derived including polynomial solutions and rational solutions by applying the unified method. The polynomial solutions include solitary wave solutions, soliton wave solutions and elliptic wave solutions, and the rational solutions contain periodic type rational solutions and soliton type rational solutions. By means of the improved F-expansion method with Riccati equation, the hyperbolic trigonometric solutions, trigonometric solutions and rational solutions are obtained containing several free parameters. The modified Kudryashov method is also applied to obtain new traveling wave solutions. In addition, the properties of several solutions are represented graphically with the appropriate parameters. [ABSTRACT FROM AUTHOR]

• Optical solutions are investigated to the (2 + 1)-dimensional chiral NLS equation with the absence and the presence of the Atangana derivative via the unified method. • The obliqueness and fractionally impacts on the optical solutions are explained. • Optical wave patterns are changed with the increase of wave obliqueness and fractionality. • The amplitudes of the optical waves are varying upon changing the constant coefficients of the model. The study aims to explore obliquely propagating optical wave solutions to the (2 + 1)-dimensional chiral nonlinear Schrödinger (NLS) equation in both the absence and presence of the Atangana derivative. In order to convert the classical-order chiral nonlinear Schrödinger equation to an ordinary differential equation, a transformation associated with wave obliqueness is applied. Hereafter, the unified method is applied to the reduced equation. As outcomes, the dark, periodic singular with unequal wave length, periodic with equal wavelength, periodic, unsmooth periodic, singular periodic soliton solutions are received to the ordinary differential equation. Later, the acquired solutions are then put to the applied transformation associated with obliqueness. Moreover, the fractional-order chiral NLS equation is solved by using the spatiotemporal Atangana derivative with oblique wave transformation and the unified method. In terms of wave obliqueness, fractionality, and applied technique sense, all generated wave solutions are revealed to be novel. Along with their physical explanations, the impacts of obliqueness and fractionality on the solutions are graphically illustrated. It is exposed that as obliqueness and fractionality increase, the optical wave phenomena change. Additionally, it is discovered that the employed method can be used to obtain novel optical soliton features to the chiral nonlinear Schrödinger equation with or without fractional and obliqueness constraints. It can be assured that the utilized method is more powerful than the other methods. As a result, the method can be used in future research to explain the many physical phenomena that arise in optical fiber communication networks. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation is one of the significant nonlinear complex models describing the optical solitons in dispersive media. In this study, the (2+1)-dimensional chiral nonlinear Schrödinger model has been investigated analytically which is of key importantance in the field of fluid sciences. A variety of exclusive travelling waveform solutions have been established for the complex dynamical model by employing a set of eminent analytical approaches namely the extended modified auxiliary equation mapping method, the improved F -expansion method, and the unified method, respectively. We obtained different forms of solitary types solutions with the help of these technique. The outcomes are a set of bell-shaped, single periodic, optical, and multi-periodic solutions. Finally, the stability of the developed results is also established to validate the computations. The study provides a very spectacular and appropriate way to put together several interesting wave demonstrations for more complex models of current era. [ABSTRACT FROM AUTHOR]

Over the last twenty years, several “different” hyperbolic tangent function methods have been proposed to search solutions for nonlinear partial differential equations (NPDEs). The most common of these methods were the tanh-function method, the extended tanh-function method, the modified extended tanh-function method, and the complex tanh-function method. Besides the excellent sides of these methods, weaknesses and deficiencies of each method were encountered. The authors realized that they did not actually give “very different and comprehensive results”, and some of them are even unnecessary. Therefore, these methods were analysed and significant findings obtained. Firstly, they compared all of these methods with each other and gave the connections between them; and secondly, they proposed a more general method to obtain many more solutions for NPDEs, some of which having never been obtained before, and thus to overcome weaknesses and deficiencies of existing hyperbolic tangent function methods in the literature. This new method, named as the unified method, provides many more solutions in a straightforward, concise and elegant manner without reproducing a lot of different forms of the same solution. Lastly, they demonstrate the effectiveness of the unifed tanh method by seeking more exact solutions of the Rabinovich wave equation which were not obtained before.

In this paper, we show that the unified method is not only more general than the family of tanh function methods, also it gives many more general solutions than the new members of the family of (G'/G) -expansion methods. Compared to other methods, the significant contribution of the unified method is firstly to unify the family of tanh function methods and the family of (G'/G) -expansion methods. Secondly, it gives many more solutions for NPDEs direct, concise and simple manner than the total of these two families. Also, the unified method gives these abundant solutions without using tedious and complex algorithm on computer programs. Afterwards, we demonstrate the effectiveness of the unified tanh method by seeking more exact solutions of the Lonngren wave equation. [ABSTRACT FROM AUTHOR]

In this work, we construct rational and double-soliton rational solutions of the KdV-Sawada-Kotera-Ramani equation with variable coefficients by using the unified method and its generalized form. We employ these methods to obtain soliton rational solutions, periodic rational solutions, elliptic rational solutions, and two-soliton rational solutions. Here, we study the nonlinear interactions between these solutions and the collision between the long surface water waves. Also, we discuss the dynamical behavior of the traveling wave solutions and their structures. [ABSTRACT FROM AUTHOR]

This study is used to investigate the exact explicit solutions and dynamical behaviors of the (1+1)-dimensional integrable system of Drinfel'd–Sokolov–Wilson equations in dispersive media. Firstly, the unified method is implemented to find explicit solutions in polynomial and rational forms. These solutions include rational, dark and bright soliton structures. After that, the planar dynamical system of the considered model is obtained using the Galilean transformation. The phase portraits of the bifurcations are drawn from the planar dynamical system for different physical parametric values using the fourth-order Runge–Kutta method. Further, an external force is imposed on the dynamical system to observe the phase portraits of chaotic trajectories. These tracks are outlined for different values of the strength and frequency of the external force acting on the dynamical system. The outcomes show that a given model has chaotic behvior as its solution becomes disordered by taking small variation in the strength and frequency of the external force. The results are new and can be helpful in investigating the dynamical behaviors of the other nonlinear physical models arising in optics, bio-mathematics and so many other fields of science. • Soliton structures and dynamical behavior. • Drinfel'd–Sokolov–Wilson equations in dispersive media. • Fourth-order Runge–Kutta method. [ABSTRACT FROM AUTHOR]

• Spatiotemporal dynamics of a diffusive predator-prey (PP) system are investigated via the unified method. • Kink, anti-kink, singular, periodic-singular, and sinusoidal wave solutions are attained. • The predator and prey population densities are changed with the change in the free parameters. • Wave solutions to the fractional diffusive PP system are reported through Atangana conformable derivative sense. • The periodicity and smoothness increase for fractional wave solutions as the value of the fractional parameter increases. This study investigates some analytic solutions and phase portraits to the diffusive predator-prey system in studying the spatiotemporal dynamics of a predator-prey community in ecology through an analytical approach and a qualitative theory of planar dynamical systems, respectively. To accomplish such aims, a simple wave transformation is applied to the diffusive predator-prey system for converting it into a system of ordinary differential equations with a planar dynamical system for analyzing the behavior of bifurcation properties and analytic solutions. Then, the analytical unified method is employed to the attained system. The applied wave transformation is put back to the obtained solutions of the system of ordinary differential equations. Finally, the analytic solutions namely, kink, anti-kink, singular, periodic-singular, and sinusoidal wave solutions are attained to the considered system. All the constructed wave solutions are found to be new from the viewpoint of the application of the unified method. In order to verify the biological wave phenomena of predator and prey populations, some graphs are presented for illustrating the analytic solutions, which have interesting implications in ecology. The effects of free parameters and wave celerity on the attained solutions are demonstrated graphically along with their physical descriptions. The graphical outputs reveal that the predator and prey population densities are changed with the change in the free parameters. Wave solutions to the fractional diffusive predator-prey system are also reported with Atangana conformable derivative sense. As the value of the fractional parameter increases, the smoothness of the anti-kink wave profile is found to increase gradually, but the steepness decreases. For the sinusoidal wave profile, the periodicity and smoothness increase as the value of the fractional parameter increases. Thus, the present study may enrich the interpretation of the spatiotemporal dynamics of predator-prey interaction in a real environment. It is also found that the applied method and the relevant transformation are effective and easy to use for acquiring new analytic solutions to the diffusive predator-prey system over the other analytic methods. The method of interest is novel and efficient because it overcomes the weaknesses and deficiencies of the other analytic methods. Therefore, the method can be applied to further studies to explain various physical phenomena arising in ecology. [ABSTRACT FROM AUTHOR]