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The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt ( ) model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model's stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines. [ABSTRACT FROM AUTHOR]

This study employs precise, accurate, and efficient analytical and numerical techniques to obtain novel and accurate solitary wave solutions for the (2+1)-dimensional Ablowitz–Kaup–Newell–Segur (AKNS) equation. The Khater II (Khat II) method and extended cubic-B-spline (ECBS) scheme are utilized as recent computational and accurate numerical techniques, respectively, for investigating the AKNS equation and constructing highly precise solitary wave solutions. The stability characteristics of the obtained solutions are also analyzed to ensure their applicability. These solutions are characterized by their peak amplitudes, wave speeds, and widths. Through comparisons with previous numerical techniques, our proposed approach demonstrates superior effectiveness and reliability. The research is significant due to the application of the (2+1)-dimensional AKNS equation in modeling various physical phenomena, such as nonlinear optics, water waves, and plasma physics. The precise and efficient determination of solitary wave solutions not only enhances the understanding of these phenomena but also provides a valuable framework for practical applications. The results have substantial implications for comprehending complex wave dynamics in physical systems and serve as a valuable tool for further research and analysis in the field of wave dynamics. [ABSTRACT FROM AUTHOR]

This investigation employs advanced computational techniques to ascertain novel and precise solitary wave solutions of the Camassa–Holm ( ℋ) equation, a partial differential equation governing wave phenomena in one-dimensional media. Originally designed for the representation of shallow water waves, the ℋ equation has exhibited versatility across various disciplines, including nonlinear optics and elasticity theory. It intricately delineates the interplay between nonlinear and dispersive effects in wave systems, with nonlinearity arising from component interactions and dispersion rooted in the temporal spreading of waves. Furthermore, the ℋ equation governs the spatiotemporal evolution of wave profiles, encompassing both nonlinear and dispersive influences. Notably, the equation allows for soliton solutions — localized wave packets sustaining their form over extended distances. The identification of precise solitary wave solutions holds paramount significance for comprehending the ℋ equation's behavior in diverse physical contexts, such as fluid dynamics and nonlinear optics. Moreover, this study establishes a correlation between the investigated model and plasma physics, demonstrating the efficacy and efficiency of the employed computational techniques through benchmarking against alternative computational methods. This augmentation underscores the broader relevance of the ℋ equation, extending its applicability to provide insights into wave phenomena analogous to those encountered in plasma physics. [ABSTRACT FROM AUTHOR]

This study investigates some novel solitary wave solutions of the perturbed Chen–Lee–Liu ( ℒ ℒ) equation to explain the physical and dynamic behavior of the pulse in optical fiber. The perturbed ℒ ℒ equation is one of the icon models that have been derived from the well-known Schrödinger equation. Two analytical techniques are used to obtain novel solitary wave solutions. Then, using a well-known numerical scheme, these solutions are looked at objectively to figure out why they work. Several graphs in different styles are used to show how the pulse waves in an optical fiber are analyzed and how accurate the analysis is. The scientific novelty of this research is given by explaining the comparison figure out of our results that have been published in some recently published research papers. [ABSTRACT FROM AUTHOR]

In this investigation, we utilize two recent analytical schemes to unveil novel solitary wave solutions for the (1 + 1) -dimensional Mikhailov–Novikov–Wang integrable equation. The said equation serves as a mathematical model that captures specific physical phenomena, albeit lacking a direct physical interpretation. Nevertheless, it finds relevance in various systems within the realm of nonlinear waves in physics. Through the application of the aforementioned analytical schemes, we derive fresh solutions and evaluate their accuracy by employing the variational iteration method. The congruence observed between the analytical and numerical solutions of the investigated model serves as validation for the constructed solutions. Furthermore, we delve into exploring the implications of obtaining precise and ground breaking solitary wave solutions on the practical applications associated with the studied model. [ABSTRACT FROM AUTHOR]

This study explores the novel solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation, aiming to elucidate the physical and dynamic behaviors of pulses in optical fiber. The perturbed CLL equation is derived from the well-known Schrödinger equation and serves as an iconic model. Two analytical techniques are employed to obtain these novel solitary wave solutions. Subsequently, these solutions are subjected to objective analysis using a widely recognized semianalytical scheme to comprehend their underlying mechanisms. Multiple graphs with diverse styles are utilized to illustrate the analysis of pulse waves in optical fiber and assess the accuracy of the analysis. The scientific novelty of this research lies in providing a comprehensive explanation through a comparative analysis of our recently published results in related research papers. [ABSTRACT FROM AUTHOR]

This study focuses on employing recent and accurate computational techniques, specifically the Sardar-sub equation () method, to explore novel solitary wave solutions of the Gilson–Pickering (ℙ) equation. The GP equation is a mathematical model with implications in fluid dynamics and wave phenomena. It describes the behavior of solitary waves, which are localized disturbances propagating through a medium without changing shape. The physical significance of the ℙ equation lies in its ability to capture the dynamics of solitary waves in various systems, including water waves, optical fibers, and nonlinear acoustic waves. The study's findings contribute to the advancement of mathematical modeling approaches and offer valuable insights into solitary wave phenomena. The stability of the constructed solutions is investigated using the properties of the Hamiltonian system. The accuracy of the computational solutions is demonstrated by comparing them with approximate solutions obtained through He's variational iteration (ℍ ) method. Furthermore, the effectiveness of the employed computational techniques is validated through comparisons with other existing methods. [ABSTRACT FROM AUTHOR]

In recent years, there has been growing interest in fractional differential equations, which extend the concept of ordinary differential equations by including fractional-order derivatives. The fractional Chaffee–Infante ( ℂ ) equation, a nonlinear partial differential equation that describes physical systems with fractional-order dynamics, has received particular attention. Previous studies have explored analytical solutions for this equation using the method of solitary wave solutions, which seeks traveling wave solutions that are localized in space and time. To construct these solutions, the extended Khater II ( ℍ ) method was used in conjunction with the properties of the truncated Mittag-Leffler ( ) function. The resulting soliton wave solutions demonstrate how solitary waves propagate through the system and can be used to investigate the system's response to different stimuli. The accuracy of the solutions is verified using the variational iteration () technique. This study demonstrates the effectiveness of analytical and numerical methods for finding accurate solitary wave solutions to the ℂ equation, and how these methods can be used to gain insights into the behavior of physical systems with fractional-order dynamics. [ABSTRACT FROM AUTHOR]

This study investigates novel solitary wave solutions of the Gilson–Pickering ( ℙ) equation, which is a model that describes the motion of a fluid in a porous medium. An analytical scheme is applied to construct these solutions, utilizing the extended Khater method in conjunction with the homogenous balance technique. The derived expressions for the solitary wave solutions are exact and are presented in terms of hyperbolic functions. The ℙ equation is valuable for a wide range of applications, including oil and gas reservoir engineering, groundwater flow, and flow in biological tissues. Additionally, this model is employed to describe the behavior of waves in various physical systems such as fluids and plasmas. Specifically, it models the propagation of dispersive waves in a media that exhibits both dispersion and dissipation. To ensure the accuracy of the constructed solutions, a numerical scheme is employed. The properties of the solitary wave solutions are analyzed, and their physical implications are explored. The results of this investigation reveal a rich variety of solitary wave solutions that exhibit interesting behaviors, including oscillatory and non-oscillatory behavior, which are elucidated through various types of distinct graphs. Consequently, this study provides significant insights into the behavior of fluid flow in porous media and its applications in various fields, including oil and gas reservoir engineering and groundwater flow modeling. The analytical and numerical methods employed in this investigation demonstrate their potential for studying nonlinear evolution equations and their applications in the physical sciences. [ABSTRACT FROM AUTHOR]

In this paper, the generalized exponential (GExp) method has been employed to construct novel solitary wave solutions of the nonlinear fractional biological population (FBP) model. This model is used to demonstrate the relation of the population with deaths and births. Many novel traveling wave solutions have been formulated in distinct forms such as exponential, hyperbolic and trigonometric forms. These solutions have been explained in three different axes. The first axis is plotting them in their three optional (real, imaginary and absolute value), the second axis is handling these solutions for constructing the requested conditions for applying trigonometric quintic B-spline (TQBS) scheme. The second one determines the accuracy of the obtained analytical solutions by showing the error's value between the analytical and numerical solutions. At the same time, the third one is comparing our analytical and numerical solutions, which have recently been published that explain the paper's contribution and novelty. [ABSTRACT FROM AUTHOR]

This study introduces recent computational advancements targeting the discovery of precise and novel solitary wave solutions in the (1 + 1)-dimensional Mikhailov-Novikov-Wang (MNW ) integrable equation. Solitary wave solutions are of paramount importance for comprehending complex nonlinear phenomena and have practical implications in diverse scientific fields such as fluid dynamics, nonlinear optics, and plasma physics. The proposed computational techniques encompass powerful numerical methods, including the Khater II (KHA - II) method, modified Rational method (MRE ) method, and He’s variational iteration (HVI ) method. These techniques offer robust and accurate approaches for computing solitary wave solutions by leveraging the inherent integrability properties of the MNW equation. Notably, these approaches demonstrate substantial improvements in efficiency, accuracy, and the ability to explore a broader range of solitary wave solutions compared to traditional computational methods. The efficacy of these techniques is effectively demonstrated through extensive numerical simulations and comprehensive comparisons with other computational approaches. The results highlight the superior performance of the proposed methods in identifying new solitary wave solutions, precisely characterizing their properties, and providing deeper insights into the model’s behavior. Overall, these computational techniques significantly contribute to advancing our understanding of the underlying dynamics of the MNW equation and hold promising potential for applications across multiple scientific disciplines. [ABSTRACT FROM AUTHOR]

In this paper, the solitary wave solutions of the complex perturbed Gerdjikov–Ivanov (CPGI) equation are investigated by employing the extended simplest equation (ESE) and modified Khater (MKhat) methods. The studied equation describes the physical characterization of the optical soliton waves to mitigate internet bottlenecks with many different applications in the telecommunication industry. The evaluated solutions are explained through various sketches in two-, three-dimensional, and contour plots of their real, imaginary, and absolute values. The novelty of the obtained results is demonstrated by comparing them with the previously obtained solutions. The computational applied schemes' performance is tested to illustrate their powerful and effective handling of many nonlinear evolutions equations. [ABSTRACT FROM AUTHOR]

This paper examines three different recent computational schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, and modified Khater (MKha) method) for obtaining novel solitary wave solutions of cubic–quintic nonlinear Helmholtz (CQ–NLH) model. This model is considered as a general model of the well-known Schrödinger equation where it takes into account the effects of backward scattering that are neglected in the more common nonlinear Schrödinger model. Many distinct wave solutions are explained in the different formulas, such as trigonometric, rational, and hyperbolic formulas. These solutions are described in some precise sketches in two- and three-dimensional. The methods' performance is explained to demonstrate their effectiveness and power. [ABSTRACT FROM AUTHOR]

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified. [ABSTRACT FROM AUTHOR]

In this paper, the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods are employed to find the soliton, breather, kink, periodic kink, and lump wave solutions of the Ostrovsky equation. This model is considered as a mathematical modification model of the Korteweg-de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. Many distinct solutions are obtained through the employed computational schemes. Moreover, some solutions are sketched in 2D, 3D, and contour plots. The effective and powerful of the two used computational schemes are tested. Furthermore, the accuracy of the obtained solutions is examined through a comparison between them and that had been obtained in previously published research. [ABSTRACT FROM AUTHOR]

In this survey, the ionic current along microtubules equation is handled by applying the modified Khater method to get the solitary wave solutions that describe the ionic transport throughout the intracellular environment which describes the behavior of many applications in a biological non-linear dispatch line for ionic currents. The obtained solutions support many researchers who are concerned with the discussion of the physical properties of the ionic currents along microtubules. Microtubules are one of the main components of the cytoskeleton, and function in many operations, comprehensive constitutional backing, intracellular transmit, and DNA division. Moreover, we also study the stability property of our obtained solutions. All obtained solutions are verified by backing them into the original equation by using MAPLE 18 and MATHEMATICA 11.2. These solutions show the power and effective of the used method and its ability for applying to many other different forms of non-linear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper studies (2+1)-dimensional Konopelchenko–Dubrovsky equation and (2+1)-dimensional KdV equation via a modified auxiliary equation technique. These two systems describe the connection between the nonlinear weaves with a weak scattering and long-range interactions between the tropical, mid-latitude troposphere, the interaction of equatorial and mid-latitude Rossby waves, respectively. We implement a novel technique to these systems to find analytical traveling wave solutions. The performance of this novel method shows its ability for applying on various nonlinear partial differential equations. All solutions obtained are checked by the Maple software system and verified for its high fidelity. [ABSTRACT FROM AUTHOR]

This manuscript handles the nonlinear Klein–Fock–Gordon (KFG) equation by applying two recent computational schemes (generalized exponential function (GEF) and generalized Riccati expansion (GRE) methods) to construct abundant novel wave solutions The considered model is the generalized form of the well-known nonlinear Schrödinger equation which is considered a quantized version of the relativistic energy-momentum relation. The accuracy of the employed analytical schemes by showing the matching between computational and approximate solutions and calculating the absolute value of error between these solutions. This matching is investigated by employing the variational iteration (VI) method to show the precision of the used schemes with the previously published solutions. The physical characterization of the evaluated solutions has explained through some distinct sketches in 2D, 3D, contour, polar, and spherical plots. The originality and novelty of our investigation have been checked by comparing our solution's accuracy with previous other solutions' accuracy. [ABSTRACT FROM AUTHOR]