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This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation. [ABSTRACT FROM AUTHOR]

This research paper investigates the behavior of optical fiber pulses by studying the solitary wave solutions of the perturbed Chen–Lee–Liu (CLL) equation. The perturbed CLL equation is derived from the perturbed Schrödinger equation. Two analytical approaches for obtaining the innovative soliton-wave solutions are presented in this paper, and their reliability is examined by using a well-established numerical method. The accuracy of pulse wave analysis in an optical cable is demonstrated through a series of graphical representations. Our study introduces scientific novelty, as evidenced by the comparative analysis of our data with those of previous research papers. This research contributes to enhancing the understanding of the kinetics and physical behavior of pulses in optical fibers, which holds implications for the advancement of optical communication technologies. [ABSTRACT FROM AUTHOR]

This research delves into the Schäfer–Wayne equation, a theoretical model that describes the nonlinear propagation of ultra-short pulses in silica optical fibers. The significance of this study lies in our utilization of three distinct analytical techniques to obtain a range of solutions. Additionally, we employ the multiplier approach to calculate conservation laws and identify the Lie point symmetries of the equation. A novel aspect of our investigation is the exploration of periodic, quasiperiodic, and weakly chaotic behaviors of nonlinear optical pulses for the Schäfer–Wayne equation, a topic explored here for the first time. To elaborate on our findings, we provide detailed descriptions of specific solutions. This work contributes to our understanding of the nonlinear dynamics of optical pulses and extends beyond previous efforts in the literature by introducing new analytical methods and uncovering unexplored behaviors within the context of the Schäfer–Wayne equation. [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 study undertakes the challenging task of unraveling the intricacies embedded in the integrable Kuralay equation, a cornerstone in the field of mathematical physics. Specifically, our focus extends to the integrable dynamics governing space curves, encompassing both the Kuralay-I equation ( K - IE ) and the Kuralay-II equation ( K - IIE ). Our methodological approach leverages innovative analytical techniques, namely the Bernoulli sub-equation function (BSF ) method and the Khater II ( K h I I ) method. For the derivation of numerical solutions, we employ the exponential cubic-B-spline (ECBS ) scheme. The outcomes of our investigation signify a triumphant resolution of the Kuralay equation, thereby enriching our comprehension of its dynamic behavior. The profound impact of this study reverberates throughout the domain of mathematical physics, particularly in illuminating the intricate dynamics inherent in integrable systems. The distinctive aspect of our research lies in the application of the BSF and K h I I methods for analytical solutions, coupled with the utilization of the ECBS scheme for numerical solutions. These methodologies, hitherto underexplored in the context of Kuralay equations, impart a unique quality to our investigation. In conclusion, this research constitutes a significant contribution to the progression of mathematical physics. It not only furnishes valuable insights, numerical solutions, and innovative methodologies for addressing the complex Kuralay equation but also advances our collective understanding of this foundational problem. [ABSTRACT FROM AUTHOR]

This research provides the first known solution to the nonlinear Hamiltonian amplitude equation using the combined Bernoulli sub-equation function and trigonometric-quintic-B-spline methods. While previous works have analyzed this model through limited linear approximations, an exact solution accounting for nonlinearity remains lacking. We address this gap by employing innovative techniques to reveal and scrutinize solitary wave solutions, elucidating complex behaviors inaccessible through linear methods. Our findings show the unequivocal effectiveness of our approach, comprising the first successful solution to this equation. This breakthrough fundamentally advances the comprehension of nonlinear physical systems across domains encompassing wave propagation, quantum mechanics, and optical phenomena. [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 research paper investigates the numerical solutions of the nonlinear fractional Ostrovsky equation through five recent numerical schemes (Adomian decomposition (AD), El Kalla (EK), Cubic B-Spline (CBS), extended Cubic B-Spline (ECBS), exponential Cubic B-Spline (ExCBS) schemes). We investigate the obtained computational solutions via the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods. 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 show the localized wave packet as a persistent and dominant feature. The accuracy of the obtained numerical solutions is investigated by calculating the absolute error between the exact and numerical solutions. Many sketches are given to illustrate the matching between the exact and numerical solutions. [ABSTRACT FROM AUTHOR]

In this study, the generalized formula of the Hirota–Satsuma coupled KdV equation derived by Hirota and Satsuma in 1981 [Hirota and Satsuma, Phys. Lett. A 85, 407−408 (1981)] is analytically and semi-analytically investigated. This model is formulated to describe the interaction of two long undulations with diverse dispersion relations; that is why it is also known with a generalized model of the well-known KdV equation. The generalized Kudryashov and Adomian decomposition methods construct novel soliton wave and semi-analytical solutions. These solutions are represented in some distinct graphs to show the waves' interactions. In addition, the accuracy of solutions is verified by comparing the obtained analytical and semi-analytical solutions that show the matching between them. All solutions are checked by putting them back into the original model through Mathematica 12. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the interaction of the rational function, hyperbolic as well as exponential function for the (3+1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation with the aid of Hirota's bilinear approach. We find several families of lump-type solutions. This method is a powerful and advantageous mathematical tool for establishing abundant lump solutions of nonlinear partial differential equations. In order to illustrate their dynamic properties, some figures are plotted with determined parameters. [ABSTRACT FROM AUTHOR]

In this survey, the ionic currents along the microtubules equation handled by applying two different techniques. It describes the ionic transport throughout the intracellular environment. This phenomenon explains the behavior of many applications in a biological nonlinear dispatch line for ionic currents. Using two different analytical techniques on this equation allows us to get new forms of analytical solitary traveling wave solutions. The obtained solutions support many researchers 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. The thing which makes us applies this property in many applications for this model. The earned results demonstrate that the proposed methods are significantly powerful, evangelist, relaxing, suitable, and convenient for solving many nonlinear models. [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]

This research paper investigates novel explicit wave solutions of the fractional Korteweg–de Vries (KdV) equation and the fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony (ZKBBM) equation. These models are used as gravity models in water and an interaction model between the long waves. The Atangana–Baleanu ( ℬ ℛ) fractional operator is utilized for the first time to convert the fractional form of both models into nonlinear partial differential equations with an integer order. The extended simplest equation method is employed to construct some distinct types of solitary wave solutions such as exponential, rational, hyperbolic and trigonometric functions. For more illustration of our obtained solutions, some figures for them are given. The power and practical properties of the used method are tested. [ABSTRACT FROM AUTHOR]

In this paper, the stable analytical solutions' accuracy of the nonlinear fractional nonlinear time–space telegraph (FNLTST) equation is investigated along with applying the trigonometric-quantic-B-spline (TQBS) method. This investigation depends on using the obtained analytical solutions to get the initial and boundary conditions that allow applying the numerical scheme in an easy and smooth way. Additionally, this paper aims to investigate the accuracy of the obtained analytical solutions after checking their stable property through using the properties of the Hamiltonian system. The considered model for this study is formulated by Oliver Heaviside in 1880 to define the advanced or voltage spectrum of electrified transmission, with day-to-day distances from the electrified communication or the application of electromagnetic waves. The matching between the analytical and numerical solutions is explained by some distinct sketches such as two-dimensional, scatter matrix, distribution, spline connected, bar normal, filling with two colors plots. [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]

The accuracy of analytical obtained solutions of the fractional nonlinear space–time telegraph equation that has been constructed in (Hamed and Khater in J. Math., 2020) is checked through five recent semi-analytical and numerical techniques. Adomian decomposition (AD), El Kalla (EK), cubic B-spline (CBS), extended cubic B-spline (ECBS), and exponential cubic B-spline (ExCBS) schemes are used to explain the matching between analytical and approximate solutions, which shows the accuracy of constructed traveling wave solutions. In 1880, Oliver Heaviside derived the considered model to describe the cutting-edge or voltage of an electrified transmission. The matching between solutions has been explained by plotting them in some different sketches. [ABSTRACT FROM AUTHOR]

This manuscript employs the trigonometric Quintic B‐spline (TQBS) scheme for investigating the numerical solution of the conformable fractional nonlinear time‐space telegraph equation. This model is derived by Oliver Heaviside 1880 to describe the cutting‐edge or voltage of an electrified transmission range with the day yet distance from an electrified transmission or electromagnetic wave's application. In Hamed and Khater, three recent computational schemes (sech–tanh expansion method; extended sinh–Gorden expansion method; and extended simplest equation method) have been applied to the fractional model for constructing the exact traveling wave solutions. Many distinct solutions have been obtained, and some of them have been sketched in two, three‐dimensional, and density plots. Here, these solutions are investigated to evaluate the initial and boundary conditions that apply the suggested numerical scheme. The accuracy of the constructed exact solutions is investigated by calculating the absolute value of error. The most accurate numerical schemes of the three applied computational schemes have been illustrated. [ABSTRACT FROM AUTHOR]

Fractional derivative is nonlocal, which is more suitable to simulate physical phenomena and provides more accurate models of physical systems such as earthquake vibration and polymers. The present study suggested a new numerical approach for the fractional diffusion-wave equation (FDWE). The fractional-order derivative is in the Riemann-Liouville (R-L) sense. Discussed the theoretical analysis of stability, consistency, and convergence. The numerical examples demonstrate that the method is more workable and excellently holds the theoretical analysis, showing the scheme's feasibility. [ABSTRACT FROM AUTHOR]

Fractional derivative is nonlocal, which is more suitable to simulate physical phenomena and provides more accurate models of physical systems such as earthquake vibration and polymers. The present study suggested a new numerical approach for the fractional diffusion-wave equation (FDWE). The fractional-order derivative is in the Riemann-Liouville (R-L) sense. Discussed the theoretical analysis of stability, consistency, and convergence. The numerical examples demonstrate that the method is more workable and excellently holds the theoretical analysis, showing the scheme's feasibility. [ABSTRACT FROM AUTHOR]

In this research, a particular attention has been given on the fractional Fokas-Lenells equation via Atangana-Baleanu derivative operator that describe the propagation of short pulses in optical fibers. The integrability properties has been investigated while using the modified Khater method that we present in details. As a result, number of new soliton solutions are obtained along with constraints on some parameters that are introduced accordingly. For further details about our obtained solutions, some distinct types of solutions have been illustrated to explain more physical and dynamical behavior of the short wave pulses in the optical fibers. [ABSTRACT FROM AUTHOR]

In this research paper, our work is connected with one of the most popular models in quantum magnetoplasma applications. The computational wave and numerical solutions of the Atangana conformable derivative (1 + 3)-Zakharov-Kuznetsov (ZK) equation with power-law nonlinearity are investigated via the modified Khater method and septic-B-spline scheme. This model is formulated and derived by employing the well-known reductive perturbation method. Applying the modified Khater (mK) method, septic B-spline scheme to the (1 + 3)-ZK equation with power-law nonlinearity after harnessing suitable wave transformation gives plentiful unprecedented ion-solitary wave solutions. Stability property is checked for our results to show their applicability for applying in the model's applications. The result solutions are constructed along with their 2D, 3D, and contour graphical configurations for clarity and exactitude. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical, semianalytical, and numerical solutions of the 2 + 1 –dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation. The extended simplest equation method, the sech-tanh method, the Adomian decomposition method, and cubic spline scheme are employed to obtain distinct formulas of solitary waves that are employed to calculate the initial and boundary conditions. Consequently, the numerical solutions of this model can be investigated. Moreover, their stability properties are also analyzed. The solutions obtained by means of these techniques are compared to unravel relations between them and their characteristics illustrated under the suitable choice of the parameter values. [ABSTRACT FROM AUTHOR]

In this work, the researchers for computing an exact solution of the (2 + 1) dimensions Kundu–Mukherjee–Naskar (KMN) equation used a newly developed technique, namely, the new extended direct algebraic method. This powerful method gives an exact solution to the KMN equation considered in this paper. The results are new and attest to the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This research paper investigates the computational solutions of the resonant schrödinger's equation. The modified Khater method and Adomian decomposition method are applyied for construct new analytical traveling and semi-analytical wave solutions. This model describes the pulse phenomena and studied in non-linear optics. For further illustration of our obtained solutions, some distinct types of sketches are given. [ABSTRACT FROM AUTHOR]

New analytical wave solutions are investigated of the fractional nonlinear (2 + 1) -dimensional Boiti–Leon–Manna–Pempinelli equation to explain more physical properties of the dynamical behavior of the incompressible fluid. A new fractional definition is used through the modified Khater method to get novel solitary wave solutions of this model. The stability property of the obtained solutions is tested to show the ability of our obtained solutions in using through the physical experiments. The novelty and advantage of the proposed method are illustrated by applying to this model. Some sketches are plotted to show more about the dynamical performance of this model. [ABSTRACT FROM AUTHOR]

This research paper employs two different computational schemes to the couple Boiti–Leon–Pempinelli system and the (3+1)-dimensional Kadomtsev–Petviashvili equation to find novel explicit wave solutions for these models. Both models depict a generalized form of the dispersive long wave equation. The complex, exponential, hyperbolic, and trigonometric function solutions are some of the obtained solutions by using the modified Khater method and the Jacobi elliptical function method. Moreover, their stability properties are also analyzed, and for more interpretation of the physical features of the obtained solutions, some sketches are plotted. Additionally, the novelty of our paper is explained by displaying the similarity and difference between the obtained solutions and those obtained in a different research paper. The performance of both methods is tested to show their ability to be applied to several nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This research paper applies the modified Khater method and the generalized Kudryashov method to the general Degasperis–Procesi (DP) equation, which is used to describe the dynamical behavior of the shallow water outflows. Some shock peakon wave solutions are obtained by using these methods. Moreover, some figures are sketched for these solutions to explain more physical properties of the general DP equation and to figure out the coincidence between different types of obtained solutions. The stability property by using the features of the Hamiltonian system is tested to some obtained solutions to show their ability for applying in the model's applications. The obtained solutions were verified with Maple 16 & Mathematica 12 by placing them back into the original equations. The performance of these methods shows their power and effectiveness for applying to many different forms of the nonlinear evolution equations with an integer and fractional order. [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 the analytical and semi-analytical wave solutions for the longitudinal wave equation. Moreover, it examines the performance of the modified auxiliary equation method and Adomian decomposition method on this model. This model describes the dispersion in the circular rod that dispersion caused by the transverse Poisson's effect in electro-magneto-elastic. Many explicit wave solutions are found by using the analytical technique. These solutions allow studying the physical properties of this model. The comparison between the analytical and semi-analytical solutions is discussed to show the value of the absolute error between them. [ABSTRACT FROM AUTHOR]

This paper studies the analytical and numerical solutions for (2+1)-D Konopelchenko- Dubrovsky equation. It also examines the performance of the modified simplest equation method and the cubic B-spline scheme on this model. Many explicit wave solutions are found by using the analytical technique. These solutions allow studying the physical properties of this model. The comparison between the analytical and numerical solutions are discussed to show which one of cubic B-spline scheme families is more accurate in finding the numerical solutions of this model. [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]

The analytical and numerical solutions of the (2+1) dimensional, Fisher-Kolmogorov-Petrovskii-Piskunov ((2+1) D-Fisher-KPP) model are investigated by employing the modified direct algebraic (MDA), modified Kudryashov (MKud.), and trigonometric-quantic B-spline (TQBS) schemes. This model, which arises in population genetics and nematic liquid crystals, describes the reaction–diffusion system by traveling waves in population genetics and the propagation of domain walls, pattern formation in bi-stable systems, and nematic liquid crystals. Many novel analytical solutions are constructed. These solutions are used to evaluate the requested numerical technique's conditions. The numerical solutions of the considered model are studied, and the absolute value of error between analytical and numerical is calculated to demonstrate the matching between both solutions. Some figures are represented to explain the obtained analytical solutions and the match between analytical and numerical results. The used schemes' performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

The dynamical attitude of the transmission for the nerve impulses of a nervous system, which is mathematically formulated by the Atangana–Baleanu (AB) time-fractional FitzHugh–Nagumo (FN) equation, is computationally and numerically investigated via two distinct schemes. These schemes are the improved Riccati expansion method and B-spline schemes. Additionally, the stability behavior of the analytical evaluated solutions is illustrated based on the characteristics of the Hamiltonian to explain the applicability of them in the model's applications. Also, the physical and dynamical behaviors of the gained solutions are clarified by sketching them in three different types of plots. The practical side and power of applied methods are shown to explain their ability to use on many other nonlinear evaluation equations. [ABSTRACT FROM AUTHOR]

In this research, we examine exact and solitary traveling solutions of nonlinear complex Kundu-Eckhaus equation with variable coefficients. Nonlinear complex Kundu-Eckhaus equation model has vital role in description of the proliferation of the pulses of the ultra-short femto second in an optical fiber, weakly nonlinear fragmented water waves, nonlinear optics and the quantum field theory. We implement a new auxiliary equation method and novel G′G-expansion technique on this model to obtain a new exact and solitary traveling wave solutions. The propose of this research is applying new method on this models to discover new form of soliton solutions that this makes the study of the electric and magnetic field of these pulses and the extent of their effect in the optical fiber more interesting. [ABSTRACT FROM AUTHOR]