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This study explores solutions for a mathematical equation called the time-space fractional phi-four equation using two methods: the Sardar-subequation method and the modified extended auxiliary equation method. The phi-four equation is connected to the Klein–Gordon model and is important in different scientific areas like biology and nuclear physics. Understanding its solutions is crucial. By using a specific wave transformation, the equation is changed into a simpler form for analysis. The methods proposed give a variety of solutions, such as Kink, bright singular, dark, combo dark bright, periodic, and singular periodic solutions. Each solution we find using these methods has specific rules that determine when it's correct. We carefully choose specific values for the parameters to help us understand more about the solutions. This helps us see the detailed features of the solutions and improves our understanding of how the model behaves in the real world. These methods create a strong framework for studying solitons, which are specific types of mathematical solutions. The study compares the outcomes of these methods with earlier ones to get a complete understanding. Graphical illustrations are used to visually represent some of these solutions, helping us grasp their characteristics. Visual representations in two- and three-dimensional figures add originality to the findings. Importantly, these methods can be applied to solve similar problems with fractional derivatives in various scientific contexts. In summary, this research not only deepens our understanding of the phi-four equation but also introduces powerful methods with broad applications in fractional differential equations. [ABSTRACT FROM AUTHOR]

In this work, the Hirota bilinear symbolic computational method along with test functions and the generalized exponential rational function method are capitalized to secure soliton and lump solutions to the Sharma–Tasso–Olver–Burgers equation. Several novel soliton solutions are observed in unique patterns such as periodic, exponential, hyperbolic, dark, singular, and combo forms. Additionally, we also extract interaction, lump and breather solutions of the governing model. The novel characteristic of this work is the attained results, which were not before computed. Modulation instability of the governing equation is also examined via linear stability theory. To demonstrate the physical aspects and configuration of the attained solitons, some distinct graphs are plotted in different shapes. The validity of the solutions is verified by using Mathematica. The constructed outcomes are very encouraging and entail that the concerned methods can be utilized to acquire assorted improved, innovative, and advantageous outcomes for miscellaneous remarkable nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

The primary aim of this paper is to investigate the dynamic behavior of generalized nonlinear fractional Tzitzéica-type equations and to derive optical soliton solutions. To achieve this goal, we employ the modified Khater method, focusing on obtaining solitary wave solutions for generalized fractional Tzitzéica-type (TT) equations. Through this approach, we unveil novel solutions for both Tzitzéica and Tzitzéica–Dodd–Bullough (TDB) equations expressed in terms of fractional derivatives. The significance of employing the modified Khater method lies in its ability to yield a diverse array of soliton solutions. These solutions encompass dark, bright, singular, periodic, kink, singular kink, and combined dark–bright solitons. The derived solutions are visually represented through two-dimensional (2D) and three-dimensional (3D) graphs. Our findings underscore that the proposed method serves as a comprehensive and efficient approach to explore exact solitary wave solutions for generalized fractional TT evolution equations. By employing the modified Khater method, we not only enhance our understanding of the dynamic behavior of these equations, but also provide a versatile tool for obtaining precise soliton solutions in the realm of nonlinear fractional evolution equations. [ABSTRACT FROM AUTHOR]

The concern of this study is to investigate the optical solitons pluses. The (2 + 1) -dimensional Kundu–Mukherjee–Naskar (KMN) mathematical model in birefringent fibers is taken under consideration for this sake because this model has great importance in optics and delineate the propagation of soliton dynamics in optical fiber communication system and the rogue waves in the oceans and the bending of the light beam. Two newly developed approaches first extended rational sinh-Gordon method which is formulated from the well-known sinh-Gordon equation and second exp (− ϕ (ϖ)) -expansion function method are manipulated for the construction of optical solitons in distinct formats. Bright, dark, and singular along their combined forms, periodic and plane wave solutions are extracted with the aid of proposed methods. Moreover, the modulation instability of investigated model is also carried out via linear stability theory. To endorse the physical compatibility of the results, the 2D, 3D, contour, and density plots have been delineated using appropriate parametric values. According to our literature research, these two methods that we are working on have not been applied to the KMN equation in birefringent fibers before, and we believe that the new solutions we have obtained will be useful to researchers working in modeling in this field. The evaluated results suggested that the techniques employed in this research to recover inclusive and standard solutions are approachable, efficient, and speedier in computing and can be considered a handy tool in solving more complex phenomena with the assistance of symbolic computation. [ABSTRACT FROM AUTHOR]

This study explores optical soliton solutions within the framework of the Caudrey–Dodd–Gibbon equation (CDGE) and the ( 1 + 1 )-dimensional Boussinesq equation, utilizing the modified Sardar sub-equation method (MSSEM). The derived solutions are meticulously verified using symbolic software Mathematica and encompass a rich array of mathematical functions, including trigonometric, hyperbolic, and exponential functions. Visualization techniques, such as 3D plots, 2D plots, density graphs, and contour graphs, effectively illustrate the diverse behaviors exhibited by these soliton solutions. These equations have been studied previously, their comprehensive traveling wave solutions are relatively unexplored. By employing the modified Sardar sub-equation method, which has demonstrated effectiveness in solving nonlinear evolution equations, this study aims to bridge this gap. The soliton solutions obtained encompass a wide spectrum of behaviors, including bright, dark, periodic, singular, dark-periodic, bright-periodic, dark-singular, compactons, and bright-singular soliton solutions, as well as other complex phenomena. The comprehensive exploration of soliton solutions not only enhances our understanding of fundamental wave phenomena but also provides valuable insights for addressing nonlinear equations across various scientific disciplines, including optical fiber communications, plasma physics, and nonlinear dynamics. [ABSTRACT FROM AUTHOR]

This research extensively investigates the fractional coupled Konno–Oono model, a prevalent framework in diverse scientific and engineering disciplines. The primary objective is to unravel the intricate dynamics embedded in this model. We adeptly convert partial differential equations into ordinary differential equation by applying appropriate transformations. Through the innovative approaches of the modified Khater method and the modified extended auxiliary equation method, we systematically derive solutions, encompassing periodic, dark, bright, singular, kink, and anti-kink solitons. Notably, these outcomes hold significant applications in the realm of magnetic fields. To assess the system's stability, we employ the linear stability mechanism, offering insights into modulation instability gain. Careful selection of parametric values allows us to conduct numerical simulations, presenting results through visually engaging figures that facilitate a deeper understanding of their significance. Importantly, our findings affirm that the applied techniques not only yield complete and uniform responses but also showcase simplicity, effectiveness, and remarkable computational efficiency. It is crucial to highlight that all the results presented in this study are obtained using the Mathematica software, emphasizing the reliability and precision of our computational approach. Overall, our research contributes valuable insights into the behavior of the fractional coupled Konno–Oono model, paving the way for enhanced comprehension and potential applications in magnetic field studies. [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]

This article investigates the truncated time M-fractional paraxial wave equation. This model is frequently used to depict the activation of waves in utterly different physical frameworks, such as quantum mechanics and optics. Two trustworthy methodologies, the improved F-expansion and modified exponent function method, are used to obtain the different soliton solutions to the truncated time M-fractional paraxial wave equation. The obtained solutions offer a valuable understanding of the underlying physical events. Discussion is also had over the equation’s modulation instability, which confirms the given equation is stable. Several graphical charts, including 2D, 3D, density, and contour, are produced using symbolic software. These visual representations have a significant positive impact on qualitative evaluations of diverse natural occurrences. The evaluated findings showed that the approaches employed in this work to obtain inclusive and standard solutions are efficient and speedier in computing, they will be beneficial in addressing more difficult higher order nonlinear perturbed truncated time M-fractional models. [ABSTRACT FROM AUTHOR]

The primary objective of this study is to extract nonlinear wave patterns from the (4+1)-dimensional Boiti–Leon–Manna–Pempinelli (4D-BLMP) equation, considering both constant and time-dependent coefficients, which is used widely to describe the incompressible fluid. By employing the amended extended tanh-function method, we successfully obtained innovative solutions in the form of combo dark bright, hyperbolic or lumps, periodic, and singular mix solitons solutions, and others. To ensure the utmost precision and reliability of our findings, we rigorously confirm them using the robust Mathematica software. These solutions hold paramount importance in the domains of in the study of incompressible fluids and acoustic waves, enriching our understanding of the foundational physical principles embedded within the equation. The study visually presents the computed wave solutions using 2D, 3D, and contour plots, effectively representing the internal structure of the phenomenon. This study proves that the computational method used is efficient, brief, and widely applicable, making it valuable to engineers who work with engineering models and dynamical models. This research can help to better understand physical phenomena in many areas of applied physics, particularly in the study of incompressible fluids and acoustic waves. [ABSTRACT FROM AUTHOR]

This article examines the nonlinear (2+1) complex conformable time fractional nonlinear Schrödinger equation and the soliton solutions that may be found by using the improved F-expansion method. Many novel solutions of concatenated model such as periodic wave, dark soliton, singular, hyperbolic, trigonometric and rational wave soliton solutions are retrieved using proposed method. The modulation instability of the selected model through stability analysis is also discussed. In order to display the retrieved soliton solutions graphically, 2D, 3D, density, and contour graphs have been utilized. The retrieved soliton solutions play a significant role in nonlinear optics. The results prove that the suggested approach is a very straightforward, concise and dynamic addition in literature. Also, these results are novel and provide invaluable insight into the fundamental physical phenomena. [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]

The article is focused on the analysis of a mathematical model known as the Nizhnik–Novikov–Veselov (SNNV) system, which incorporates a specialized mathematical tool called the truncated M-fractional derivative (TMD). This model has broad applications in various scientific fields and considered an isotropic Lax extension of the one-dimensional Kdv equation which has diverse applications in the areas of warm ions, shallow water waves, electromagnetic signals, and river irrigation flows. To extract the solutions of the SNNV system, a modified version of the extended tanh-function method which is known as amended extended tanh-function method (AETF) is brought into play to derive solitary wave solutions for the SNNV system. These formulated solutions are categorized as trigonometric, exponential, and hyperbolic functions. For physical exemplification and knowledge of the dynamic physical features of the governing model, some reported solutions have been deliberated graphically by selecting appropriate parametric values. The computed outcomes have implications in diverse areas of applied sciences such as quantum mechanics, magneto-electrodynamics, heavy ion collisions, optical fiber technology and also opens doors to practical applications in diverse domains. The application of the AETF approach in this study demonstrates its efficiency and competence in scrutinizing and extracting soliton solutions in nonlinear partial differential equation. [ABSTRACT FROM AUTHOR]

Numerous variations of cognitive challenges, such as those in fluid mechanics, plasma physics and nonlinear optics as well as in engineering and mathematics, involve nonlinear partial differential equations. In this study, we explore the (3+1)-dimensional generalized Yu–Toda–Sasa–Fukuyama (YTSF) equation with application in engineering and physical science. Three (1+1)-dimensional nonlinear partial differential equations can be acquired from the YTSF equation. Using a N-fold Darboux transformation technique of Lax pair to obtain the multi-soliton, resonant and complex soliton solutions of the equation. Also, by showing the solutions graphically, the completeness of the outcome was confirmed. The conclusions in this study might be useful for understanding the soliton solutions in mathematics and physics. [ABSTRACT FROM AUTHOR]

The stochastic nonlinear Schrödinger model (SNLSM) in (1+1)-dimension with random potential is examined in this paper. The analysis of the evolution of nonlinear dispersive waves in a totally disordered medium depends heavily on the model under investigation. This study has three main objectives. Firstly, for the SNLSM, derive stochastic precise solutions by using the modified Sardar sub-equation technique. This technique is efficient and intuitive for solving such models, as shown by the generated solutions, which can be described as trigonometric, hyperbolic, bright, single and dark. Secondly, for obtaining numerical solutions to the SNLSM, the algorithms described here offer an accurate and efficient technique. Lastly, investigate the phase plane analysis of the perturbed and unperturbed dynamical system and the time series analysis of the governing model. The results show that the numerical and analytical techniques can be extended to solve other nonlinear partial differential equations in physics and engineering. The results of this study have a significant impact on how well we comprehend how solitons behave in physical systems. Additionally, they may serve as a foundation for the development of improved numerical techniques for handling challenging nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

The article focuses on exploring three distinct equations: the Jimbo-Miwa equation (JME), the generalized shallow water equation (GSWE), and the Hirota-Satsuma-Ito equation (HSIE). By applying the exp (- Φ (η)) -expansion method (EEM), we have successfully obtained novel solutions with trigonometric, elliptic, and hyperbolic properties. The main objective of this study is to identify and explore previously undiscovered soliton solutions within nonlinear wave equations, contributing to a deeper comprehension of wave behaviors and facilitating potential applications across diverse scientific and engineering domains. The Jimbo-Miwa equation is relevant to integrable systems and mathematical physics, potentially finding applications in quantum field theory and condensed matter physics. The generalized shallow water equation extends the classical shallow water equations, enabling better modeling of complex fluid dynamics like ocean currents and tsunamis. The Hirota-Satsuma-Ito equation, likely a soliton-based nonlinear equation, holds importance in nonlinear optics, fluid dynamics, and possibly biological studies, contributing to the comprehension of wave-like behaviors in diverse systems. Soliton and solitary wave structures are extracted as distinct solutions. By selecting appropriate values for arbitrary parameters within the accurate range, we create 3D, 2D, and contour plots to visualize the discovered solutions. Modifying model parameters enables the alteration of the solution dynamics generated by the models. The calculations for this research were exclusively performed using the symbolic software Mathematica. The solutions received encompass a variety of types, such as dark, bright, combo dark-bright, singular, cuspons, peakons, periodic solitary wave solutions, single-soliton solutions, double-soliton solutions, N-soliton solutions, and numerous others. These solutions have real-life applications in areas such as predicting coastal hazards, improving optical communications, studying nonlinear dynamics, enhancing material science, and advancing medical imaging techniques. The complexity and nonlinear nature of the system are underscored by these findings, emphasizing the necessity for additional analysis. Moreover, the obtained results offer valuable insights into understanding and modeling comparable physical systems. This research marks a significant advancement by utilizing the the exp (- Φ (η)) -expansion method to reveal solitonic solutions for an unsolved model, thereby expanding the existing literature and introducing a novel mathematical technique to address nonlinear physical models. The proposed method is concise, transparent, and reliable, leading to reduced computations and widespread applicability. [ABSTRACT FROM AUTHOR]

This study is devoted to investigating the soliton solutions of the (2+1) dimensional truncated M-fractional Heisenberg ferromagnetic spin chain model. The model has applications in modern magnetism theory and it illustrates the feature of magnetism to many insulating crystals as well as interaction spins. By deploying two analytical approaches, the improved F-expansion method and the modified exponential function method we document different form soliton solutions such as dark, exponential, periodic, and plane wave solutions. The stability of the studied model is also carried out with the assistance of linear stability which affirms that the model is stable. To understand the solitonic dynamics of the attained solutions, various graphs are sketched by assigning appropriate values to parameters. The predictable accomplished soliton solutions will offer a rich plinth to study the nonlinear spin dynamics in magnetic materials. The computed soliton solutions are novel compared with those achieved previously by other investigators who used diverse methods which justify the originality of the work. The calculated solutions confirm the techniques' competence, applicability, capability and can be handy tools to handle more intriguing higher-order nonlinear perturbed models. [ABSTRACT FROM AUTHOR]

In this article, we examine the Kuramoto–Sivashinsky equation, a nonlinear model that modifies a variety of physical and chemical scenarios. The main goal of this article is to find the analytical solution to the fuzzy fractional Kuramoto–Sivashinsky equations (FFKSEs). We employ the fractional reduced differential transform method (FRDTM) for dealing with the fractional Caputo operator. By using three examples, we tested the developed technique. The upper and lower parts of the fuzzy solutions for all three challenges were obtained using two different fractional-order simulations between 0 and 1. The attained series-type result was identified in the current investigation. The results show that the suggested strategy leads to accurate solutions at the integer level. The outcomes of applying the suggested methodology demonstrate the effectiveness and simplicity of our approach for analysing the behaviour of nonlinear models used in science and technology. [ABSTRACT FROM AUTHOR]

In this research, we address the problem of solving (1 + 1)-dimensional fractional coupled nonlinear Schrödinger equations (FCNLSE) with beta derivatives, which are essential for understanding wave dynamics in various physical systems. These equations have significant importance in practical applications, particularly in the design of optical fiber networks, signal processing, and control systems, where precise modeling of wave behavior is crucial. To tackle this problem, we employ two powerful mathematical methodologies, the modified exponential function method and the rational exp( - φ (η ))-expansion method. These methods are known for their ability to provide accurate analytical solutions to fractional nonlinear physical models, making them invaluable tools for solving complex mathematical problems. The growing popularity of fractional nonlinear partial differential equations stems from their versatile applicability, which extends to diverse domains of science and engineering. To approach the FCNLSE problem, we leverage a well-suited fractional complex wave transformation, effectively translating the original equation into a more tractable ordinary differential equation. This transformation sets the stage for the discovery of a wide range of solutions that encompass compactons, periodic cross-kink structures, peakons, as well as rational and cuspons solutions. These solutions are expressed in terms of rational, hyperbolic, trigonometric, and exponential functions, providing a rich mathematical tapestry to analyze and interpret. To enhance our comprehension of the physical significance of these solutions, we employ advanced visualization techniques, including the generation of three-dimensional, two-dimensional, and contour plots. These graphical representations offer a vivid insight into the dynamic behavior of the obtained solutions. Our findings not only emphasize the precision and effectiveness of the applied methodologies but also contribute significantly to the understanding of various physical phenomena. These novel solutions extend beyond previous efforts in the literature by introducing beta derivatives as a modeling tool for FCNLSE. Additionally, we uncover solution types that have not been previously reported, such as periodic cross-kink structures, expanding the landscape of possible solutions for FCNLSE. [ABSTRACT FROM AUTHOR]

This paper reveals soliton solutions to ( 4 + 1 )-dimensional Fokas equation, which is an integrable extension of the Kadomtsev–Petviashvili (KP) and Davey–Stewartson (DS) equations. In wave theory, the Fokas equation plays a crucial role in describing the physical phenomena of waves on the water's surface and beneath it. The observed model is subjected to an extended simple equation method (ESEM) and the Hirota bilinear method (HBM), which disclosed an abundance of soliton solutions in distinct formats, in the form of trigonometric functions, singular, periodic, rational, and exponential solutions. Moreover, we developed a number of solutions, such as the homoclinic breather wave solution,the periodic wave solution, the M-shaped rational wave solution, and the kink with their interaction solution, which are not documented in the literature. Additionally, modulation instability is effectively discussed. Some of the achieved results are explained in 2D, 3D, contour and density graphs. The new results interpreting that these obtained solutions can be a part, to complete the family of solutions and considered methods are effective, simple, and easy to use. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the impact of the integrability criterion on mixed-derivative nonlinear Schrödinger equations, specifically focusing on the Rangwala-Rao (R R) equation introduced by A. Rangwala in 1990. Our objective is to enhance our understanding of the dispersion effect by examining innovative soliton wave solutions and their interactions. The tanh method is utilized to generate unique solitary wave solutions for the Rangwala-Rao (R R) equation. The study of the Rangwala-Rao (R R) holds significance as it has the potential to contribute to the development of more efficient optical fiber communication systems. The numerical solutions presented in this research illustrate the dynamic nature of optical fiber pulse propagation, underscoring the distinctiveness of this work compared to previous scholarly endeavors. The study obtained various soliton solutions, including rational, trigonometric, and hyperbolic functions. The results are presented in graphical form with appropriate parameter values to aid visualization. The originality of our computed outcomes, which represent novel achievements that surpass previously derived solutions, becomes evident when we compare our accomplishments with theirs. These newly examined results are innovative and novel, with the potential to significantly propel the realms of nonlinear optics and mathematical physics forward. This study proves that the computational method used is efficient, brief, and widely applicable, making it valuable to engineers who work with engineering models and dynamical models. [ABSTRACT FROM AUTHOR]

The fractional coupled nonlinear Schrödinger model (FCNLSM) is widely utilized in various fields such as nonlinear optics, optical communication systems, plasmas and mathematical physics. In this study, we aim to achieve three primary objectives. Firstly, we seek to obtain novel soliton solutions for the FCNLSM, which have not been previously reported in the literature. Secondly, we employ the Sardar sub-equation method and the improved generalized tanh-function method to effectively analyze the dynamics of solutions and solve the studied model. These methods provide valuable insights into the behavior of the system. Lastly, we conduct bifurcation and stability analyses to explore the dynamical properties of the model. To ensure the physical validity of our findings, we present 2-dimensional, 3-dimensional and contour plots using carefully selected parameter values. The obtained results demonstrate the feasibility, efficiency and computational speed of the employed techniques in obtaining comprehensive and reliable solutions. The study represents a novel and significant contribution to the field by expanding the understanding of soliton solutions in the FCNLSM, introducing new techniques for their investigation and conducting a comprehensive analysis of bifurcation and stability properties. The findings of this research open new avenues for exploration and application in the areas of nonlinear optics, optical communication systems and other related fields. [ABSTRACT FROM AUTHOR]

This study investigates the fractionally coupled nonlinear Schrödinger model (FCNLSM), which has numerous applications in different fields of physics, such as optics, condensed matter physics and plasma physics. The study employs two versatile techniques, the unified technique and the modified F -expansion technique, to explore various solutions. By applying these techniques, we obtain novel soliton solutions, which are expressed in terms of rational, hyperbolic and trigonometric solutions, along with kink, periodic and singular soliton solutions. Additionally, multi-wave U-shaped solitary wave solutions are assessed. The sensitivity analysis of the model is investigated and distinctive 2-dimensional, 3-dimensional and density graphs are used to illustrate the behavioral characteristics of the retrieved solutions. As far as we know, this manner of investigation has never been explored before. The results demonstrate the reliability, consistency and effectiveness in finding precise solutions to the various difficult nonlinear issues that arise in engineering, applied sciences and nonlinear optics. [ABSTRACT FROM AUTHOR]

The current research analyses several new optical solitons solutions to the Lakshmanan–Porsezian–Daniel (LPD) equation with the Kerr law of nonlinearity which arises from the application of the Heisenberg spin chain and fiber optics, via two modified analytical methodologies. The auxiliary equation method (AE) and improved F-expansion method are used to derive several accurate solitary wave solutions of the LPD model. The key objectives of the study are in two folds. Firstly, the implementation of these schemes yields dark, bright, multi-wave, singular, alphabetic and exotic solitons solutions to the model. Secondly, we discuss the stability analysis of our selected model which confirms that the governing model is stable. For understanding the physical configuration of some constructed solitons, we plotted 2D, 3D, contours and density graphs. These solutions secured in this study will play a vital role to understand the exact dynamics of nonlinear phenomena in nonlinear optics and also enhance the previously established results. In the present era of communications network technology and nonlinear optics, the employed techniques seem to be a more categorical, influential and proficient approaches for securing optical solutions to several diversified contemporary models. [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]

In this research, we study traveling wave solutions to the fractional extended nonlinear SchrÖdinger equation (NLSE), and the effects of the third-order dispersion parameter. This equation is used to simulate the propagation of femtosecond, plasma physic and in nonlinear optical fiber. To accomplish this goal, we use the extended simple equation approach and the improved F-expansion method to secure a variety of distinct solutions in the form of dark, singular, periodic, rational, and exponential waves. Also, the stability of the outcomes is effectively examined. Several graphs have been sketched under appropriate parametric values to reinforce some reported findings. Computational work along with a graphical demonstration confirms the exactness of the proposed methods. The issue has not previously been investigated by taking into account the impact of the third order dispersion parameter. The main objective of this study is to obtain the different kinds of traveling wave solutions of fractional extended NLSE which are absent in the literature which justify the novelty of this study. We believe that these novel solutions hold a prominent place in the fields of nonlinear sciences and optical engineering because these solutions will enables a through understanding of the development and dynamic nature of such models. The obtained results indicate the reliability, efficiency, and capability of the implemented technique to determine wide-spectral stable traveling wave solutions to nonlinear equations emerging in various branches of scientific, technological, and engineering domains. [ABSTRACT FROM AUTHOR]

The main focus of this study is to extract the optical solitons of the two variants of the nonlinear Schrödinger model including the dispersive cubic-quintic nonlinear Schrödinger equation and nonlinear Schrödinger equation with group velocity dispersion. These models have dynamic applications in diversified domains of applied sciences, optical engineering and can be used in the propagation of light in nonlinear optical fibers. For this purpose, a well-organized method called the exp -function method is implemented and derives optical soliton in the shape of dark, bright, periodic and kink. To validate the physical compatibility of the results, the 2D, 3D, contour, and density plots have been outlined using appropriate parametric values which assist the investigators to comprehend the physical phenomena of the governing equation. The results established in this manuscript are fresh and extend the numbers of previously published results. The acquired solutions exhibit that the applied computational methodology used is brief, concise, and reliable, resulting in fewer computations and broad applicability. The scrutinized wave's outcomes are trustworthy to the researchers and also have imperious applications in mathematics and optical physics. [ABSTRACT FROM AUTHOR]

In this paper, we consider the conformable resonant nonlinear Schrödinger equation (CRNLSE) incorporating Kerr law nonlinearity and provide some new analytical solutions. Three analytical methods such as the Exp-function method, modified exp (- Φ (η)) -expansion function method, and exp (- Φ (η)) -expansion function method are utilized to handle this problem. These methods yield various innovative and fascinating solutions, including trigonometric, exponential, periodic, bright, singular, dark, and rational solutions, as well as their constraint conditions. Moreover the modulation instability (MI) of the consider model is also investigated. For each solution discovered, 2D, 3D, and contour plots are also sketched to clarify their physical configuration. The reported solutions enhance the previously established results. The solutions demonstrate that these methods are efficient and effective for locating traveling wave solutions and could be a beneficial tool for handling other complex nonlinear partial differential equations (NLPDEs) emerge in diversified scientific fields such as hydrodynamics, nonlinear optics, nonlinear fibers and plasmas. [ABSTRACT FROM AUTHOR]

The perturbed nonlinear Schrödinger equation is used frequently to simulate ultra-short pulse lasers, nonlinear optics, optical communication systems, plasmas and other areas of mathematical physics and engineering. The main objective of this study is two fold. (1) to obtain the different kinds of soliton solutions of perturbed nonlinear Schrödinger equation with kerr law nonlinearity which are absent in the literature, (2) the implementation of improved F-expansion method for such studied model with Modulation Instability. As far as our best knowledge, this have never been studied before in this way. To endorse the physical compatibility of the results, the 2D, 3D, contour, and density plots have been delineated using appropriate parametric values. The evaluated results suggested that the technique employed in this research to recover inclusive and standard solutions is approachable, efficient, and speedier in computing and can be considered a handy tool in solving more complex phenomena that arise in engineering, mathematical physics and optical fiber. [ABSTRACT FROM AUTHOR]

This study deals to investigate a diverse collection of optical solitons and others solutions to fiber Bragg gratings with dispersive reflectivity having Kerr law of nonlinear refractive index. Bragg gratings are no doubt a technological spectacle that sustained balance between dispersion and nonlinear effects that leads to a stable transmission of solitons across transnational distances. Two recently established norms extended sine–cosine/sinh–cosh and generalized exponential rational function method are capitalizing to formulate bright, dark, singular, combo, complex and exponential solutions. The stability analysis of the studied model is also examined by utilizing linear stability theory. By selecting suitable parametric values, the dynamics of the evaluated results are exemplified by sketching their 2D, 3D and contour profiles. The novelty of the gained outcomes is manifested by a detailed comparison with the results that already exist. The applied computational approaches are concise, outspoken and reliable which lessen the tedious computational work and give it a wide range of applicability. [ABSTRACT FROM AUTHOR]

In an elastic rod the longitudinal deformation wave propagation is modeled by the nonlinear partial differential equation known as the Pochhammer-Chree equation. In this article, a conformable fractional order generalized Pochhammer-Chree equation with the n order term is studied for constructing some new analytical solutions by using a proficient analytical technique. The solitary wave solutions are established by using the Exp-function method for the presented model equation describing the longitudinal vibration of the material in a thin, straight cylindrical rod. By considering the different parameter conditions, the existence of different kinds of solitary wave solutions is determined which are also presented in the form of 3D plots, contour plots, and 2D plots to visualize and explicate the physical structure of the problem. [ABSTRACT FROM AUTHOR]

In this present study, the modified generalized exponential rational function method (mGERFM) and the Hirota bilinear method (HBM) are implemented to secure optical and lump solutions to the fractional coupled nonlinear Schrödinger equations (FCNSE). These problems are an appealing model to describe the modes in nonlinear optics and Bose-Einstein condensation. Numerous novel soliton solutions are computed in distinct formats such as periodic, exponential, dark and combo singular bright. In addition, we also evaluate multi waves, periodic cross-kink, rational, and interaction solutions along with rational, trigonometric, and various bilinear functions. The novel feature of this study is the acquired solutions, which were not before constructed and signify a good balance between the nonlinear physical components. The dynamical property of the retrieve solutions is also depicted through some distinct graphs in 2-, 3-dimensional. The constructed outcomes are auspicious, which present that the stated methods are categorical, robust, and efficient in finding exact solutions to diverse complex nonlinear problems arising in the recent era of nonlinear optics, applied sciences and engineering. [ABSTRACT FROM AUTHOR]

In science and technology, the phenomena of transportation are crucial. Advection and diffusion can occur in a wide range of applications. Distinct types of decay rates are feasible for different non-equilibrium systems over lengthy periods of time when it comes to diffusion. In engineering, biology, and ecology, the problems under study are used to represent spatial impacts. The fast Adomian decomposition method (FADM) is used to solve time fractional reaction diffusion equations, which are models of physical phenomena, in the current study. Caputo fractional derivative meaning applies to the specified time derivative. The results are in series form and correspond to the proposed fractional order problem. These models have a strong physical foundation, and their numerical treatments have significant theoretical and practical applications. The leaning of the rapid convergence of method-formulated sequences towards the appropriate solution is also graphically depicted. With less computational cost, this solution quickly converged to the exact solution. [ABSTRACT FROM AUTHOR]

This article explores the application of the reduced differential transform method (RDTM) for the computational solutions of two fractional-order cancer tumor models in the Caputo sense: the model based on cancer chemotherapeutic effects which explain the relation between chemotherapeutic drugs, tumor cells, normal cells, and immune cells using a fractional partial differential equations, and the model that describes the different cases of killing rate K of cancer cells (the killing percentage of cancer cells K (I) is dependent on the number of cells, (II) is a function of time only, and (III) is a function of space only). The solutions are presented using Mathematica software as a convergent power series with elegantly computed terms using the suggested technique. The proposed method gives new series form results for various values of gamma. To clarify the complexity of the models, we plot the two- and three-dimensional and contour graphics of the obtained solutions at varied values of fractional-order gamma and the selected system parameters. The solutions are analyzed with fractional and reduced differential transform methods to obtain an idea of invariance regarding the computed solution of the designed mathematical model. The obtained results demonstrate the efficiency and preciseness of the proposed method to achieve a better understanding of chemotherapy effects. It is observed that chemotherapy drugs boost immunity against the specific cancer by decreasing the number of tumor cells, and the killing rate K of cancerous cells depend on the cells concentration. [ABSTRACT FROM AUTHOR]

The current article finds new soliton closed-form wave structures of the solutions of the fractional perturbed Schrödinger equation with Kerr law nonlinearity. The various kinds of solutions are accomplished by looking at a competent technique, the tanh–coth method. The nonlinear soliton wave prearrangement is analyzed and different types of soliton solutions are in the form of 3D-plots, contour plots, and 2D-plots by looking at the different values of the parameters presented to describe the propagation of traveling wave solutions. The results obtained are new and may be applicable for some physical fields, like optical fibers, plasma fluids, and bimolecular dynamical modes. The discovery of a new optical soliton could have ramifications in other photonics fields, such as nonlinear optics fibers and spectroscopy, fractal medium in the future. [ABSTRACT FROM AUTHOR]

The present article discovers the new soliton wave solutions and their propagation in nonlinear low-pass electrical transmission lines (NLETLs). Based on an innovative Exp-function method, multitype soliton solutions of nonlinear fractional evolution equations of NLETLs are established. The equation is reformulated to a fractional-order derivative by using the Jumarie operator. Some new results are also presented graphically to understand the real physical importance of the studied model equation. The physical interpretation of waves is represented in the form of three-dimensional and contour graphs to visualize the underlying dynamic behavior of these solutions for particular values of the parameters. Moreover, the attained outcomes are generally new for the considered model equation, and the results show that the used method is efficient, direct, and concise which can be used in more complex phenomena. [ABSTRACT FROM AUTHOR]

In this manuscript, we successfully apply the Hirota's bilinear method and extended sinh-Gordon equation expansion method to analyze different wave structures for the (2+1)-dimensional nonlinear Zakharov–Kuznetsov modified equal-width equation by considering the test function approaches. We secure some novel combo form bright, dark, singular breather waves, lump periodic, and two-wave solutions. Modulation instability analysis of the given model also be analyzed. Furthermore, the physical behavior of the achieved results is sketched through 3-dimensional, 2-dimensional and contour profiles with the assistance of suitable parameters. The results obtained are exceptional in comparison with previous findings in the literature. Computation packages are used to verify all of the secured results by putting them back into the original model equation. Thus our strategies through fortress of representative calculations give a functioning and intense mathematical execute for tackling complicated nonlinear wave issues. [ABSTRACT FROM AUTHOR]

In this paper, an efficient (G ′ / G , 1 / G) -expansion method is adopted to resolve a famous (2 + 1)-dimensional fractional Ablowitz–Kaup–Newell–Segur (AKNS) water wave equation for the non-conservative system that plays a significant role in understanding the wave propagation. This work addresses the physical and dynamic behavior of some new exact trigonometric, hyperbolic, and rational solitary wave solutions in the form of 3D-plots and contour plots using different measures of parameters. The obtained results show the efficiency of the proposed method for the analytical treatment of nonlinear problems in mathematics, science and engineering and may be helpful in better understanding the propagating wave dynamics in diverse situations. [ABSTRACT FROM AUTHOR]

The main purpose of this research is to inquire the new solitary wave solution of the coupled time-fractional models to validate the influence and proficiency of the planned variational iteration method (VIM) using conformable derivative definition. Applications to four demanding nonlinear problems like Hirota-Satsuma coupled KdV equations, modified Boussinesq (MB) equation, approximate long wave (ALW) equation and Drinfeld-Sokolov-Wilson (DSW) equation demonstrate the efficiency and the robustness of the method. An analysis of the consequences with effects of relevant parameters and comparison with the exact solution presented with the help of graphs tables and gives the further understanding of numerical results by others. The convergence of the method is illustrated numerical and their physical significance is discussed. [ABSTRACT FROM AUTHOR]

In real world physical applications purpose, it is complicated to acquire an exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic and therefore creating the need for the use of reliable and efficient techniques in the solution of fuzzy differential equations. The purpose of this research paper is to utilize the reliable analytic approach of homotopy perturbation Sumudu transform method for better understanding of systems of non-linear fuzzy integro-differential equations, while using the concept of fuzzy parameter in certain dynamical problems to remove the hurdles faced in numerical approach. These mathematical models are of great interest in engineering and physics. Some numerical examples are also given to demonstrate the efficiency and superiority of the method, followed by graphical representation of the comparison of exact and approximated solution by using Maple 2017. [ABSTRACT FROM AUTHOR]

In this work, a new emerging analytical techniques variational iteration method combine with Aboodh transform has been applied to find out the significant important analytical and convergent solution of some mathematical models of fractional order. These mathematical models are of great interest in engineering and physics. The derivative is in Caputo's sense. These analytical solutions are continuous that can be used to understand the physical phenomena without taking interpolation concept. The obtained solutions indicate the validity and great potential of Aboodh transform with the variational iteration method and show that the proposed method is a good scheme. Graphically, the movements of some solutions are presented at different values of fractional order. [ABSTRACT FROM AUTHOR]

In this work, four variants of time-fractional nonlinear Caputo-type Newell- Whitehead-Segel (NWS) problem are solved by fractional variational iteration method. This equation results as mathematical model of many important physical reaction-diffusion processes and pattern formation theory in diverse fields of chemical engineering, mechanical engineering, bio-engineering, biology and fluid mechanics, and the numerical treatments of these models are of great scientific significance and practical values. The received outcomes and plotted graphs express that the carried out technique is unpretentious, competent, and easily applicable for nonlinear fractional-order Partial differential equations (PDEs). [ABSTRACT FROM AUTHOR]

In this paper, we will use the exp(-Φ(η))\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(-\Phi (\eta ))$$\end{document}-expansion method to obtain the solitonic wave solution in the sense of the truncated M-fractional Schrödinger–KdV equation. The provided equation is converted into an ordinary differential equation using the appropriate wave transformation. Standard waveform shapes are determined, such as hyperbolic, exponential, dark, bright, rational, plane, and combo bright-dark soliton. We create 2D, density, and contour graphs of the solutions using consistent parametric values to examine the physical characteristics of the constructed solitons. Using Wolfram Mathematica, the newly created solutions are verified by inserting them back into the model under consideration. The suggested method and results can also be used to analyze high-order fractional models found in fields such as optics, hydrodynamics, plasma, wave theory, and others. [ABSTRACT FROM AUTHOR]

In the present work, some mathematical models of non-linear types for first order integro-differential equation and system of fractional order are analytically tackled by the fractional VIM. The fractional derivative is considered in Caputo Fabrizio sense involving an exponential function and has interesting properties. The illustrative numerical examples show that the solution approach to the classical solutions as the order of the fractional derivative increases. The graphs obtained by generalized results are also plotted by using symbolic software Maple. Comparison of our approximate solution with the exact numerical results of the problems is presented to illustrate the reliability and efficiency of the method. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show use of modified variational iteration technique for understanding fuzzy integro-differential equations. Illuminating integro differential equations with correct parameter inside numerous displaying physical issues isn't exactly simple or better to state unthinkable in genuine issues. To defeat this trouble fuzzy idea is utilized. Generally it's hard to comprehend nonlinear fuzzy IDEs numerically. [ABSTRACT FROM AUTHOR]

In this paper, we have given the procedure and implement the differential transform method (DTM) to solve some linear fractional differential equations under fuzzy initial value conditions. The derivative is in the sense of the Caputo fractional derivative. The solutions of considered model examples are computed in the form of convergent series with easily computable components. To display the strength and applicability of this proposed method few illustrative examples are solved and the obtained results show that the said approach is easy to implement. [ABSTRACT FROM AUTHOR]

In the present work, the optical soliton solutions to the 2-dimensional fractional coupled nonlinear Schrödinger model with B eta fractional derivative are found using the unified Riccati equation expansion (UREE ) approach. These solutions are essential to understand wave propagation in a range of physical domains. In real-world applications where precise wave motion is crucial, such as control systems, signal processing, and fiber optic networks, these models are crucial. We successfully employ the UREE approach and secure some novel optical solutions in the shape of singular, dark periodic and rational wave solutions. Additionally, to explore the fact that the model is highly sensitive, we discussed the time series and sensitivity analysis of the examined model. The investigations, which concentrate on the nonlinear dynamic behaviors of the solutions, are new and unexplored. These behaviors are shown in 3-D plots, contour plots and 2-D curves and descriptions of the related physical properties. Our results show that the use of the UREE approach for producing optical solutions and looking at them in fractional nonlinear models and dynamical data offers useful mathematical tools in applied mathematics and the investigation of wave motion in shallow water. [ABSTRACT FROM AUTHOR]

This paper is the witness of coupling of recently developed Aboodh transform and differential transform method. This coupling is known as Aboodh differential transform method. The proposed method is tested on some linear and nonlinear problems for suitable analytical solution and the obtained results demonstrate reliability and efficiency of the proposed method. It is observed that the presented coupling is an alternative approach to overcome the demerit of complex calculation. This new method is more efficient and easy to handle such differential equations in comparison to other methods. The results reveal the complete reliability that this method is very efficient, simple and applications hence can be extended to other problems of diverse nature. [ABSTRACT FROM AUTHOR]

Many mathematical models are contributed to give rise to of linear and nonlinear integral equations. This paper deals with the application of coupling of Laplace transform with decomposition method to solve analytically various nonlinear singular integral equations that described heat transfer. The elegant coupling gives rise to the modified versions of decomposition method which is very efficient in solving nonlinear problems of diversified nature of handling the Abel-type singular equation. It is observed that the decomposition method using Laplace transform is very efficient, easier to implements and more users friendly. Several examples are given to reconfirm the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]