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The current investigation seeks to explore the nonlinear Benney–Luke model, a governing equation pertinent to the propagation of extended waves within dispersive media, such as those observed in water waves and plasma waves, among other nonlinear wave phenomena. This model is distinguished by its incorporation of both dispersive and nonlinear effects, rendering it a valuable instrument for delving into intricate wave dynamics. Notably, it shares similarities with other prominent nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation and the Boussinesq equation, both of which have been extensively examined within the domain of water waves and associated phenomena.Employing analytical methodologies, specifically the Khater (Khat III) and modified Kudryashov (MKud) methods, the research endeavors to derive exact solutions for the Benney–Luke equation. These solutions serve to elucidate various aspects of long wave behavior in dispersive media, encompassing their propagation characteristics, stability, and potential for inter-wave interactions. Furthermore, a trigonometric-quantic-B-spline (TQBS) scheme is adopted as a numerical approach to corroborate the precision of the derived analytical solutions and demonstrate their relevance across diverse physical scenarios linked to the model under examination.The research outcomes underscore the efficacy of the analytical techniques in generating dependable solutions for the Benney–Luke model. Moreover, through the utilization of the TQBS numerical scheme, the accuracy of the derived analytical solutions is affirmed, thereby ensuring their pragmatic utility across a spectrum of physical scenarios pertinent to the investigated model. This endeavor holds significance in its contribution towards advancing the comprehension of nonlinear wave phenomena within dispersive media, with ramifications spanning multiple disciplines, including fluid dynamics, plasma physics, and nonlinear optics. The novelty of this study lies in its application of the Khater (Khat III) and modified Kudryashov (MKud) methods to the Benney–Luke equation, alongside the integration of the TQBS numerical scheme for solution validation. Positioned within the domain of nonlinear partial differential equations, this research aligns with its broader applications in physics, particularly concerning the investigation of dispersive wave phenomena.

This study delves into the exploration of the (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt (GFKDKK) system, a crucial nonlinear evolution equation governing wave motion across various physical domains. The core aim is to devise innovative analytical solutions for this system employing the Khater III technique and an enhanced Kudryashov methodology integrated with the conformable fractional derivative.The GFKDKK system holds pivotal significance in modeling and comprehending diverse scientific phenomena, ranging from fluid dynamics to plasma physics and nonlinear optics. This research endeavors to harness the potential of the Khater III technique and the updated Kudryashov approach to offer fresh analytical insights into nonlinear fractional differential equations, thus enriching the existing knowledge base in this domain.The attained solutions furnish invaluable insights into the dynamics of the system and its potential applications, thereby facilitating the development of effective numerical simulations and analytical frameworks. The study’s significance lies in its contribution to the ongoing exploration of nonlinear fractional differential equations and their practical implications across various industries.In conclusion, the research affirms the efficacy of the Khater III method and the enhanced Kudryashov technique, complemented by the conformable fractional derivative, in obtaining analytical solutions for the (3+1)-dimensional GFKDKK system. The novelty lies in the application of these methodologies to this specific system, resulting in the discovery of new analytical solutions that enhance our understanding of its behavior and characteristics.

The Kakutani–Matsuuchi (KM) equation provides a mathematical framework for studying the dynamics of extended nonlinear internal gravity waves in a stratified fluid medium, including important phenomena such as solitary wave propagation. This study explores various analytical and semi-analytical methods to obtain accurate solutions for the (1＋1)-dimensional KM model.The primary objective of this research is to derive new solitary wave and periodic wave solutions, and subsequently validate these analytical techniques by comparing them with solutions obtained through the Adomian decomposition method. The KM model governs the dynamics of nonlinear waves and exhibits significant connections with other nonlinear equations, notably the (KdV) equation. The analytical methods, particularly the Khater II and auxiliary equation sub-equation methods, lead to closed-form solutions by reducing the partial differential equation to a system of ordinary differential equations. Conversely, the Adomian method provides a semi-analytical solution expressed as an infinite series.Solitary and periodic traveling wave solutions are systematically derived using the aforementioned analytical techniques. To verify their accuracy, solutions obtained through Adomian decomposition are computed recursively and compared. The results highlight the effectiveness of the analytical methods in discovering new exact solutions, thereby enhancing our understanding of the physical phenomena described by the KM equation. These solutions contribute to a more nuanced understanding of the properties and behaviors of internal waves, thereby enriching our overall knowledge in this field. Consequently, this research further develops analytical solution techniques tailored for nonlinear wave models prevalent in fluid dynamics.

The focus of this study is to investigate novel and precise solitary wave solutions of the modified Kawahara (MK) equation using recent and accurate computational techniques. The MK equation, a nonlinear partial differential equation, has significant applications in various fields, including fluid dynamics and plasma physics The comprehension of nonlinear wave phenomena relies heavily on solitary wave solutions, making it necessary to develop accurate and efficient computational techniques for their identification. In this study, the Khater II method is employed as a computational technique, while the variational iteration method is used as a numerical scheme.The Khater II method is a powerful computational technique that has exhibited promising outcomes in solving nonlinear wave equations. It uses an analytical framework to transform the partial differential equation into a set of ordinary differential equations that are amenable to straightforward solutions, thereby enabling the construction of exact solutions. Conversely, the variational iteration method is a numerical scheme that enhances solution accuracy through iterative approximations.By using the Khater II method as a computational technique and the variational iteration method as a numerical scheme, this study identifies novel and precise solitary wave solutions of the MK equation. These solutions provide valuable insights into the dynamics and behavior of nonlinear waves across various applications. The effectiveness of the employed techniques is demonstrated by comparing them with other commonly utilized computational methods for solving the MK equation. The results indicate that the combined application of the Khater II method and the variational iteration method yields more accurate and efficient solutions. This heightened accuracy and efficiency are critical for comprehending and predicting the behavior of the studied model in real-world applications. Thus, this study underscores the significance of employing recent and accurate computational techniques, such as the Khater II method and the variational iteration method, to identify novel and precise solitary wave solutions of the MK equation. The obtained solutions contribute to an improved understanding of the model’s behavior and hold practical implications across diverse scientific and engineering domains.

This study discusses several analytical and approximate solutions to the generalized Camassa–Holm– Kadomtsev–Petviashvili (CHKP) problem in (2+1) dimensions, which have been recently discovered. The model describes the interesting shapes that liquid droplets can take due to dispersion. The process involves the movement of molecules in a liquid and the movement of particles caused by evaporation. Colloidal particles develop unique cracking patterns when a deposit dries up. A novel approach to solving solitary waves is presented, which combines the Khater II approach with generalized rationalization and Adomian decomposition. To demonstrate the contribution of this paper, several graphical representations, including two-dimensional, three-dimensional, and density plots, are used. The results are compared with previously published works. The methods presented in this study are effective, convenient, and easy to use in solving a range of partial differential equations that are nonlinear in nature. The use of these methods enables the accurate and efficient analysis of the behavior of the CHKP model, providing insights into the formation of interesting shapes of liquid droplets due to dispersion. The comprehensive analysis of the results using various graphical representations and comparison with previous studies enhances the scientific significance of the research.

This study investigates the fifth-order Korteweg–de Vries (fKdV) equation, which is a generalized form of the classical KdV equation governing weakly nonlinear waves in dispersive media. By employing analytical and numerical techniques, the Khater II and variational iteration methods are utilized to obtain accurate soliton wave solutions. The combination of these approaches ensures the reliability and precision of the analytical outcomes. Graphical representations, including two-dimensional, three-dimensional, and contour plots, visually illustrate the results. Additionally, the stability of the constructed solutions is assessed through the characterization of the Hamiltonian system, providing valuable insights into the soliton wave solutions’ stability properties. This research integrates analytical and numerical methods to explore the fKdV equation within the KdV-KP framework, offering novel soliton wave solutions and enhancing their interpretation through graphical representations while assessing their stability using the Hamiltonian system’s characterization.

This article investigates the accuracy of the obtained analytical solutions of the nonlinear fractional Ostrovsky (NLFO.) equation. The researched solutions are obtained by exploiting the well-known generalized Tanh–function (GTF.) method with the Atengana– conformable fractional (ACF.) derivative in the wave transformation. Five–numerical schemes (Adomian decomposition (AD), El Kalla (EK), cubic B-Spline (CBS), extended cubic B-Spline (ECBS), and exponential cubic B-Spline (ExCBS)) handle these solutions to check their solutions. The investigated fractional model is a general form of the KdV equation; however, the KdV equation’s solutions replace this effect with radiating inertia gravity waves. This model also describes a weak description of nonlinear ocean wave processes considering Earth rotation. The obtained analytical and numerical results are sketched through Mathematica 12 in different plot types to explain the waves’ dynamical behavior. Additionally, the computational obtained solutions’ stability is investigated. Finally, the paper’s contribution and obtained results’ novelty are demonstrated by comparing our solutions with those that have been obtained in previous scientific articles.

The aim of this paper is to incorporate modern computational and numerical methods to give analytical and approximate soliton solutions for the generalized fractional Hietarinta–type (GFH) model in (2+1) dimensions. Solitary wave solutions for the suggested model, which mathematically shows how gravity and surface tension make waves move again on the water’s surface, are identified and illustrated. The phase velocity of waves with different wavelengths is described by the difference in frequency in fluid dynamics. A new modern numerical technique is used to show how accurate the computational solution is, along with analytical solutions. Hamiltonian system features are also utilized to regulate the solutions’ stability. All findings are double-checked by reinserting them into Mathematica 13.1’s core model.

This study utilizes computational and numerical techniques to investigate the (1+1)-dimensional Mikhailov–Novikov–Wang (MNW) integrable equation, a nonlinear partial differential equation that governs the propagation of solitons in various physical systems, including plasma physics and fluid dynamics. The numerical methods proposed in this research provide an accurate and efficient means of obtaining wave solutions for the MNW equation. Through the application of these techniques, new soliton solutions are derived, expressed in terms of hyperbolic and trigonometric functions, and analyzed to determine their amplitude and shape characteristics. The results obtained demonstrate the potential of the proposed numerical methods to solve other nonlinear partial differential equations in fields such as engineering and physics. The findings of this study have significant implications for understanding the dynamics of solitons in physical systems and may facilitate the development of more effective numerical methods for solving nonlinear partial differential equations.

The modified Equal-Width (MEW) equation is a fundamental model in the realm of partial differential equations that is used to simulate one-dimensional wave propagation in nonlinear media, incorporating dispersion processes. In this study, we have employed two contemporary analytical and numerical methods to obtain exact traveling wave solutions for this model. The derived solutions have been evaluated using a numerical technique, demonstrating excellent accuracy. Various graphical representations, including a contour plot, a two-dimensional graph, and a three-dimensional graph, have been utilized to illustrate and analyze the results. Our study offers significant contributions to the field by providing new insights into the dynamics of solitary wave solutions of the MEW model. Furthermore, our method proves to be a valuable and effective mathematical tool that can be utilized to solve various nonlinear wave problems, making it highly applicable in a range of physical and mathematical studies. By showcasing the properties of the investigated model and the primary advantages of the employed techniques, our study highlights the potential for further developments and advancements in this field.

This work focuses on the accuracy and numerical strategies for solving the fractional Chaffee–Infante (CIE) equation in (2+1) dimensions computationally. This model illustrates the flow and transformation of gas as it travels through a homogeneous medium. When the constituents of a medium do not alter from their initial state, we say that the medium is homogeneous. In none of the solutions did we change the proportions of the individual components. Three novel analytical and numerical techniques provide new, dependable approaches for determining and estimating responses. The tabular data shown here facilitates interpreting the numerical data presented below. The simulations, which are exhibited in both 2D and 3D, depict the behavior of a solitary solitaire in both the natural and digital worlds. These findings demonstrate that this strategy is the most effective way to solve nonlinear mathematical physics problems.

In this article, semi-analytical and numerical simulations of the well-known modified Benjamin–Bona–Mahony (mBBM) equation are processed. This study targets to check the accuracy of the obtained analytical solutions of the mBBM model that have been obtained in [1] through three recent analytical schemes (extended simplest equation (ESE) method, modified kudryashov (mKud) method, and Sech-Tanh (ST) expansion method). The considered model describes the propagation of long waves in the nonlinear dispersive media in a visual illusion. The Homotopy iteration (HI) method, exponential cubic - B-spline (ECBS) method, and trigonometric- quantic-B-spline (TQBS) method are employed to construct novel semi-analytical and accurate numerical solutions. The obtained solutions’ accuracy has been checked through some different types of two-dimensional graphs

This article studies novel soliton wave solutions of the nonlinear fractional ill-posed Boussinesq (NLFIPB) dynamic wave equation by applying the extended Riccati-expansion (ERE) method. Jacques Hadamard has formulated the investigated model to figure out the dynamic characterizations of waves in shallow water under gravity. The obtained solutions are explained through some sketches in 2D and 3D and contour plots. At the same time, the results’ accuracy is checked by comparing the obtained solutions with semi-analytical solutions through the well–known Adomian decomposition (AD) method. The superiority of the ERE method over the original method is explained. All constructed solutions are checked by submitting them back into the original model through Mathematica 12 software.

In this study, the influence of the integrability requirement on nonlinear Schrödinger equations with mixed derivatives is examined. The Rangwala–Rao (RR) equation is named after A. Rangwala, who in 1990 was the first to quantify these effects in their entirety. The aim of our research is identifying how to generate individual waves and how they interact with one another. This is to get a better understanding of the dispersion effect and the progressive variation of the electric field envelope during pulse propagation in optical fibers. The direct algebraic method (DAM), the extended direct algebraic approach (exDAA), and the extended Riccati expansion technique (exREM) are used to develop unique solitary wave solutions for the model under consideration. These numerically calculated solutions illustrate the dynamic behavior of an optical pulse traveling through a cable. Comparing our results to those of other researchers demonstrates the originality of our work.

This study applies computational and numerical techniques to develop some novel and accurate solutions for Gross–Pitaevskii (GP) equations. A quantum system of identical bosons is described using the Hartree–Fock approximation and pseudopotential interaction model by the Gross–Pitaevskii equation (GPE), named after Eugene P. Gross and Lev Petrovich Pitaevskii. Many solitary wave solutions are constructed in various forms based on the implementation of the Khater II method and the novel Kudryashov method. The gained solutions are numerically represented in various graph styles. The validity of the solutions is investigated by applying the septic-B-spline scheme based on calculating the requested conditions from the obtained computational solutions. To make our study more applicable, we study the stability of our solutions by focusing on the Hamiltonian system’s properties. Mathematica 13.1 checks all inputs and outcomes against the original model for further confidence.

This research aims to look into alternative unique solitary wave solutions to the perturbed Chen–Lee–Liu (CLL) equation to explain the kinetic and physical aspects of an optical fiber pulse. The perturbed CLL equation is one of the most well-known icon models derived from the well-known Schrödinger equation. Two different analytical methodologies are used to develop novel solitary wave solutions. These responses are then thoroughly evaluated using a well-known numerical approach to establish their underlying veracity. Various graphics are used to show the pulse wave analysis process and outcomes in an optical cable. We show how scientifically unique the study is by explaining the comparison figure that was made from our data and has been used in many recent research studies

The Caudrey–Dodd–Gibbon (CDG) model, a variation of the fifth–order KdV equation (fKdV) with significant practical consequences, is solved in this study using a precise and numerical technique. This model shows how gravity-capillary waves, shallow–water waves driven by surface tension, and magneto-acoustic waves move through a plasma medium. With a focus on accuracy, new computational and approximation methods have been made possible by recent improvements in analytical and numerical methods. Numeric information is represented visually in the tables. All simulation results are shown in two and three dimensions to show both the numerical and fundamental behavior of the single soliton. Recent research shows that this method is the best way to solve nonlinear equations that are common in mathematical physics.

In this study, we use cutting-edge analytical and numerical approaches to the Gilson–Pickering (GP) problem in order to get precise soliton solutions. This model explains wave propagation in plasma physics and crystal lattice theory. A variety of evolution equations have been developed from the GP model, including the Fornberg–Whitham (FW) equation, the Rosenau–Hyman (RH) equation, and the Fuchssteiner-Fokas-Camassa–Holm (FFCH) equation, to name a few. The GP model has been studied using these evolution equations. To investigate the characterizations of new waves, crystal lattice theory and plasma physics use the Khater II, and He’s variational iteration approaches. Many alternative responses may be achieved by utilizing various formulae; each of these solutions is shown by a distinct graph. The validity of such methods and solutions may be demonstrated by assessing how well the relevant techniques and solutions match up. The results of this study suggest that the technique is preferred for successfully resolving nonlinear equations that emerge in mathematical physics.

In this study, we investigate the solitary wave solutions for the (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC). New solitary wave solutions are generated and checked using the Bernoulli sub-equation (BSE), a novel Kudryashov (NKud), and He’s homotopy perturbation (He’s HP). One may learn about solitary waves by looking at polar, contour, two-dimensional, or three-dimensional charts. Our research is original because it contrasts with what has already been done. Mathematica software 13.1 is utilized for checking the computational techniques solutions’ accuracy by putting them back into the original model.

This article delves into new solitary wave solutions for the unidirectional Dullin–Gottwald–Holm (DGH) model. Dynamical characterizations of wave prorogation in shallow water are investigated using two modern analytical approaches (generalized Tanh-function (GTF) methods and the improved Fan-expansion (IFE) method). In addition, the solutions achieved by using He’s variational iteration (He’s VI) approach are examined by demonstrating agreement between the built analytical and numerical solutions, providing more evidence for the correctness of the acquired results. Numerous sorts of solutions are produced, including solitary, shock, kink, periodic, cone, anti-kink, etc. The numerical simulations of these solutions are shown as 3D, 2D, and contour graphs. Some stream graphs illustrate the local direction of the vector field at each point, and a relatively uniform density across the property, indicating no background scalar field, further elucidates the interaction between solutions. In light of prior literature, the authors of this research provide an explanation for the originality of their findings. The comparison reveals up-to-date analytical and numerical solutions to the model under study.

This research studies novel analytical solutions of an Atangana conformable fractional Lotka–Volterra (LV) model system arising in ecology by means of three systematic schemes (the extended simplest equation method, modified Kudryashov method, and the sech–tanh expansion method). Then we use the obtained solutions to evaluate the boundary and initial conditions that helps to apply the B-spline collection numerical schemes to evaluate the value of absolute error. The investigation aims also to explain the accuracy of the analytical solutions by evaluating absolute value of errors between exact and numerically obtained solutions. In order to better explain the obtained solutions, some relevant sketches are given in three different types. The novelty of the work done is attempted to explain by comparing our results with some available and relevant results. Further, relevant strengths, usefulness, practical applications, and the methods’ ability for applying in various nonlinear evolution equations have been explored.

This research explores the complex and physical behavior, using four different theoretical methods, of water wave propagation with surface tension. A modern Benneye-Luke (BL) algorithm is used to identify a variety of unobtained distinct wave solution forms, such as multi-solitons, lumps and breath-solutions by modified Khater (MK) method, improved Riccati expansion (IRE) method, novel G′G expansion (NGE) method, and generalized Kudryashov (GK) method. The nature of the Hamiltonian method often discusses the properties of the model form. Furthermore, some of the solutions in various forms of plots can be clarified by the complex actions of water wave propagation.

This research is based on three main pillars which are studying the computational solutions of the modified Benjamin–Bona–Mahony (BBM) equation via the modified Khater method then investigating the stability property of the obtained solutions through the Hamiltonian system’s features, evaluating the initial and boundary conditions that allow applying the B-spline collection schemes (cubic, quantic, septic) to find the numerical solutions of the suggested model and to explain the matching between our obtained types of solutions. This model is used in the optical illusions field by describing the propagation of long waves in the nonlinear dispersive media in a visual illusion. An optical illusion is an illusion caused by the visual system and characterized. Four distinct types of sketches (2D, 3D, contour, and Stream plots) have been employed for better understanding our obtained solutions and their convergence.

In the presence of nonlinear dispersion, we examine certain solitary wave solutions of a dimensionless nonlinear Schrödinger (DLNLS) equation with the parabolic law of nonlinearity. We use a unique computational (generalized rational (GRat)) approach to build a variety of solutions with varying shapes. These answers show how the nonlinear interaction between Langmuir waves and electrons causes the parabolic law nonlinearity. For a better visual description of the examined model, these interactions are shown using several graphs in three-, two-dimensional, and polar plots. These representations of the obtained solutions are considered as numerical simulaions to explain the investigated model’s characterizations, the pulse waves’ behaviour and the interaction between complex short wave and real long wave envelope.

In this study, the numerical simulation of the fractional nonlinear third-order Schrödinger (3-FNLS) problem is processed by applying the novel homotopy perturbation (NHP) method. The 3-FNLS equation illustrates how quantum characteristics and ultra-short optical fiber pulses interact dynamically. Additionally, the wave function of a quantum mechanical system in which atomic particles act like waves is determined by this. When electrons, like light waves, travel through a double slit, they produce diffraction patterns. Consequently, it was reasonable to assume that the wave equation could effectively represent the behavior of atomic particles. The requested conditions for handling the investigated model by the NHP method through the Khater II method. Accurate approximate solutions are constructed and simulated in some distinct graphs.

The main goal of this paper is to discover some new analytical solutions of a fractional form of the Bogoyavlensky–Konopelchenko equation via two new analytical schemes. This model is considered as a particular case of (2 + 1)–dimensional version of the well–known KdV equation where it describes the interaction between the Riemann wave propagating and the long-wave propagation along the x,y–axises. An efficient fractional derivative called Atangana–Baleanu-Riemann derivative is utilized to convert the standard form of the model into a nonlinear fractional PDE with an–integer order. The basic idea in these methods is to use a new variable to transform the form of the equation into a nonlinear equation with ordinary derivatives. The novelty of the present paper is that the new solutions determined by applying these two powerful analytical methods can not be found in previous articles. Several two and three-dimensional figures have been depicted to illustrate the dynamic behavior of the acquired solutions. Another advantage of these two methods is their applicability in solving similar models using this fractional derivative operator.

In this paper, a nonlinear fractional emerging telecommunication model with higher–order dispersive cubic–quintic is studied by using two recent computational schemes. This kind of model is arising in many applications such as machine learning and deep learning, cloud computing, data science, dense sensor network, artificial intelligence convergence, integration of Internet of Things, self–service IT for business users, self-powered data centers, and dense sensor networks (DSNs) that is used in the turbine blades monitoring and health monitoring. Two practical algorithms (modified Khater method and sech–tanh functions method) are applied to higher–order dispersive cubic–quintic nonlinear complex fractional Schrödinger (NLCFS) equation. Many novel traveling wave solutions are constructed that do not exist earlier. These solutions are considered as the icon key in the emerging telecommunication field, were they able to explain the physical nature of the waves spread, especially in the dispersive medium. For more illustration, some attractive sketches are also depicted for the interpretation physically of the achieved solutions.

Novel explicit wave solutions are constructed for the fractional nonlinear model of the low–pass electrical transmission lines. A new fractional definition (Atangana–Baleanu derivative operator) is employed through the modified Khater method to get new wave solutions in distinct types 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. Moreover, the obtained analytical solutions are used to evaluate the initial and boundary conditions that allows applying the cubic & septic B–spline schemes to investigate the numerical solutions of this model. 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 behavior of this model.

In this research, analytical and numerical solutions are studied of a two–dimensional discrete electrical lattice, which is mathematically represented by the modified Zakharov–Kuznetsov equation. Moreover, the stability property of the obtained analytical solutions is investigated based on the Hamiltonian system, and then it is used to evaluate the initial and boundary conditions that are used in the numerical investigation. Many kinds of analytical solutions are obtained, such as complex, exponential, hyperbolic, and trigonometric function solutions. The functioning of both schemes of both techniques is tested and investigated.

This paper investigates the soliton wave structures of the nonlinear potential Kadomtsev–Petviashvili and Calogero–Degasperis equations by employing the direct algebraic method. These equations are used to investigate the (2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis and the stability of the one-soliton solution of the well-known Korteweg–de Vries (KdV) equation under transverse perturbations. Since the wavelength of weakly dispersive, nonlinear waves is longer than their amplitude and their variations in the second spatial dimension (rescaled y) are slower than those in the principal propagation direction (rescaled x), the studied models’ soliton wave solutions are used to describe wave dynamics. Many solutions are obtained and demonstrated by plotting them in 2D, 3D, and contour plots. The implemented analytical scheme’s performance verifies its effectiveness and power. All solutions’ accuracy is checked by putting them back into their original model.

The main goal of this study is to find solutions for the fractional model of the fifth-order weakly nonlocal Schrödinger equation incorporating nonlinearity of the parabolic law and external potential using a recent modification of the homotopy analysis method (HAM) called the q-homotopy analysis transform method (q-HATM). A mixture of q-HAM and Laplace transform is the projected solutions procedure. The method contributes approximate and exact (for some special cases) solutions such as the bright soliton, dark soliton, and exponential solutions. The simulation results using Mathematica package software, demonstrate that only a few terms are enough to achieve precise, effective, and reliable approximate solutions. In addition, in terms of plots for varying fractional order, the physical behavior of q-HATM solutions has been depicted and the numerical simulation is also exhibited. The results of q-HATM reveal that the projected method is competitive, reliable, and powerful for studying complex nonlinear models of fractional type.

The perturbed nonlinear Schrödinger–Hirota equation with spatio-temporal dispersion (PNSHE-STD) which governs the propagation of dispersive pulses in optical fibers, is investigated in this study using an improved Sardar sub-equation method. The Kerr and power laws of nonlinearity are taken into account. As a result of this improved technique, many constraint conditions required for the existence of soliton solutions emerge. We retrieved several solutions such as the bright solitons, dark solitons, singular solitons, mixed bright–dark solitons, singular-bright combo solitons, periodic, and other solutions. Furthermore, we demonstrate the dynamical behaviors and physical significance of these solutions by using different parameter values.

This paper examines the effects of three distinct numerical schemes (Adomian Decomposition, quintic & septic Spline methods) to investigate semi-analytical and approximate solutions on Wu–Zhang (ZW) system. It describes the (1+1)-dimensional dispersive long wave in two horizontal directions on shallow waters. The ZW model is one of the fractional nonlinear partial differential equations. Conformable derivatives properties are employed to convert the nonlinear fractional partial differential equation into an ordinary differential equation with integer order so as to obtain the approximate solutions for this model. The solutions obtained for each technique were compared to reveal their relationship to their characteristics illustrated under the suitable choice of the parameters values. The obtained solutions showed the power, easiness, and effectiveness of these methods on nonlinear partial differential equations. Keywords: Fractional nonlinear Wu–Zhang system, Conformable fractional derivatives, Numerical schemes

This paper studies the optical soliton solutions of a nonlinear Schrödinger equation (NLSE) involving parabolic law of nonlinearity with the presence of nonlinear dispersion by using the generalized auxiliary equation technique. As a result, new varieties of exact traveling wave solutions have been uncovered, comprising of the hyperbolic trigonometric, trigonometric, exponential, and rational. Interestingly, we obtain the bright, dark, periodic, singular, and other soliton solutions to the nonlinear model. Some of the achieved solutions are illustrated graphically in order to fully understand their physical behaviour. Furthermore, the findings discussed in this present investigation may be useful in explaining the propagation of optical solitons in a weakly nonlocal parabolic law medium.

This manuscript uses the generalized Khater (GK) method and the trigonometric quintic B-spline (TQBS) scheme to study the calculations and approximate solutions of complex nonlinear Fokas–Lenells (FL) equations. This model describes the propagation of short pulses in optical fibers. Many novel computing solutions have been obtained. The absolute, real, and imaginary values of some solutions are plotted in two three-dimensional and density graphs to explain the dynamic behavior of short pulses in the fiber. The use of constructed analytical solutions to evaluate initial and boundary conditions allows the application of numerical solutions to study the accuracy of our novel computational techniques. The performance of both methods demonstrates the ability, effectiveness, and ability to apply them to different forms of nonlinear evolution equations to check the accuracy of analytical and numerical solutions.

This paper investigates the analytical solutions of the well-known nonlinear Schrödinger (NLS) equation with the higher-order through three members of Kudryashov methods (the original Kudryashov method, modified Kudryashov method, and generalized Kudryashov method). The considered model is also known as the sub-10-fs-pulse propagation model used to describe these measurements’ implications for creating even shorter pulses. We also discuss the problem of validating these measurements. Previous measurements of such short pulses using techniques. This paper’s aim exceeds the idea of just finding the traveling wave solution of the considered model. Still, it researches to compare the used schemes’ accuracy by applying the quintic-B-Spline scheme and the convergence between three methods. Many distinct and novel solutions have been obtained and sketched, along with different techniques to show more details of the model’s dynamical behavior. Finally, the matching between analytical and numerical schemes has been shown through some tables and figures.

The Boussinesq equation simulates weakly nonlinear and long wave approximation that can be used in water waves, coastal engineering, and numerical models for water wave simulation in harbors and shallow seas. In this article, the sine-Gordon expansion (SGE) approach and the generalized Kudryashov (GK) scheme are used to establish broad-spectral solutions including unknown parameters and typical analytical solutions are recovered as a special case. The well-known bell-shape soliton, kink, singular kink, compacton, contracted bell-shape soliton, periodic soliton, anti-bell shape soliton, and other shape solitons are retrieved for the definite value of these constraints. The 3D and contour plots of some of the results obtained are sketched by assigning individual values of the parameter and analyzed the dynamical behavior of the waves. Furthermore, the compatibility of the two approaches has been compared and examined the efficiency to ascertain soliton solutions.

This manuscript investigates the accuracy of the solitary wave solutions of the (2+1)-dimensional nonlinear Chiral Schrödinger ((2+1)-D CNLS) equation that are constructed by employing two recent analytical techniques (modified Khater (MKhat) and modified Jacobian expansion (MJE) methods). This investigation is based on evaluating the initial and boundary conditions through the obtained analytical solutions then employing the Adomian decomposition (AD) method to evaluate the approximate solutions of the (2+1)-D CNLS equation. This framework gives the ability to get large complex traveling wave solutions of the considered model and shows the superiority of the employed computational schemes by comparing the absolute error for each of them. The handled model describes the edge states of the fractional quantum hall effect. Many novel solutions are obtained with various formulas such as trigonometric, rational, and hyperbolic to the studied model. For more illustration of the results, some solutions are displayed in 2D, 3D, and density plots.

The computational solutions for the fractional mathematical system form of the HIV-1 infection of CD4+ T-cells are investigated by employing three recent analytical schemes along the Atangana–Baleanu fractional (ABF) derivative. This model is affected by antiviral drug therapy, making it an accurate mathematical model to predict the evolution of dynamic population systems involving virus particles. The modified Khater (MKhat), sech–tanh expansion (STE), extended simplest equation (ESE) methods are handled the fractional system and obtained many novel solutions. The Hamiltonian system’s characterizations are used to investigate the stability property of the obtained solutions. Additionally, the solutions are sketched in two-dimensional to demonstrate a visual representation of the relationship between variables.

This manuscript investigates the analytical and semi-analytical solutions of nonlinear phi-four (PF) equation by applying the sech–tanh expansion method, modified Ψ′Ψ-expansion method and Adomian decomposition method. This equation is considered as a particular case of the well-known Klein–Fock–Gordon (KFG) equation. The KFG equation is derived by Oskar Klein and Walter Gordon and relates to Schrödinger equation. Many quantum effects can be studied based on the PF model’s solutions, such as wave-particle duality to describe reality in the form of waves is at the heart of quantum mechanics. The considered model is also used to explain de Broglie waves’ character, the spineless relativistic composite particles, relativistic electrons, etc., which are also the main icons for a good understanding of the phenomenon of quantum physics. Through two recent analytical schemes, handling this model gives many novel computational solutions that are tested through the semi-analytical scheme to investigate their accuracy. Demonstrating the obtained analytical and matching between analytical and semi-analytical through some distinct sketches shows the considered model’s novel physical properties.

This research paper investigates the computational wave solutions of the nonlinear Klein–Fock–Gordon (KFG) equation by applying two effective recent computational schemes. The nonlinear KFG model is a quantized version of the relativistic energy–momentum relation related to the well-known Schrödinger equation. The modified Khater (MK) and modified Kudryashov (MKud) computational schemes are employed to construct novel explicit wave solutions for figuring out novel physical properties of the investigated model. Some solutions are illustrated in some distinct figures. This paper’s novelty and originality are discussed by showing the similarity and differences between our paper and other published papers.

Applying several latest theoretical techniques, the empirical description of the interaction between the high-frequency Langmuir and the low-frequent ion-acoustic waves, derived mathematically by Zakharov’s non-dimensional (ZE) equation. These interactions are described in electromagnetic waves, plasma physics, signal processing through optical fibers, coastal engineering, and fluid dynamics. Three modern computing methods are being used to construct several solutions: the extended exp(ϕ)-expansion process,the popular Kudryashov process, and the modified Khather method. Centered on the properties of the Hamilton system, the stability properties of the solutions are investigated. The physical proprieties are illustrated using 3D plots. The originality of the obtained solutions is explored. We demonstrate that we retrieve the old known solutions and we obtain new solutions that have never been found before.

This research paper investigates the numerical solutions of the (1 + 1)-dimensional Ito equation through the extended simplest equation (ESE) method. This model is considered as well-known in quantum mechanics and nonlinear optics, which represents the height of the water’s free surface above a flat bottom. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. Diverse novel computational solutions are constructed and demonstrated through three distinct types of sketches. The stability of our obtained solutions is investigated by using Hamiltonian system’s characterizations. The novelty of our paper is explained by comparing our obtained solutions with the previously evaluated computational solutions with different computational schemes.

In this paper, the Zakharov (Z.) equation in the dimensionless form is numerically investigated via (Cubic & Quantic & Septic) B-spline schemes to demonstrate the fidelity of the calculated computational solutions. The Z equation depicts the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves. This interaction is expounded in the prompts of the coastal engineering, electromagnetic field, signal handling in the optical fibres, plasma physics, and fluid dynamics. Three different computational schemes were applied to the Z equation for constructing many novel analytical solutions. In our paper, we try to check the accuracy of these solutions via the above-mentioned numerical schemes. Moreover, some separate sketches are given to indicate more physical features of this interaction. The originality of the obtained solutions is investigated by showing the similarities and differences between our obtained solutions and that was purchased in previously published papers.

In this paper, we investigate distinct novel analytical and semi-analytical solutions of the higher-order nonlinear Schrödinger equation with the non-Kerr nonlinear term by the employment of there different schemes. These schemes are the generalized auxiliary equation method, the generalized exp--ϕξ expansion method, and the Adomain decomposition method that are considered as useful tools in this field. The suggested model in this study is used to explore the dynamics of light pulses for sub-10-fs-pulse propagation in the framework of computational simulations. The primary research of our study focuses on the case where the carrier wavelength of the soliton is much shorter than the spatial width. Herewith, the soliton frequency is smaller than the carrier frequency. The shorter femtosecond pulses (

In this research we apply generalized Exp-Function method to obtain exact, solitary and new soliton wave solutions of new coupled Konno-Oono equation, Higgs field equation and Maccari equation via generalized Exp-Function method which are very substantial models in define a current-fed string interacting with an external magnetic field in three-dimensional Euclidean space, introduces quantum field (or the Higgs field) to illustrate the generation mechanism of mass for gauge bosons and described the motion of the isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma physics, nonlinear optics and others. generalized Exp-Function method is very sturdy, fabulous, felicitous and effective method to get exact, solitary and new soliton wave solution of nonlinear partial differential equations (PDEs.). We present a contrasting between the results of this modern method and another method and show that how the results that obtained by this method is much closed to cover many different methods in this field and not just that but also get a new solitary and soliton wave solutions which give a wide range of solutions that help all researchers who apply these models in our life. Keywords: New coupled Konno-Oono equation, Higgs field equation, Maccari equation, Generalized Exp-Function method, Traveling wave solutions, Solitary wave solutions

In this research, we study new two techniques that called the extended simple equation method and the novel G′G-expansion method. The extended simple equation method depend on the auxiliary equation dϕdξ=α+λϕ+μϕ2 which has three ways for solving depends on the specific condition on the parameters as follow: When λ=0 this auxiliary equation reduces to Riccati equation, when α=0 this auxiliary equation reduces to Bernoulli equation and when α≠0,λ≠0,μ≠0 we the general solutions of this auxiliary equation while the novel G′G-expansion method depends also on similar auxiliary equation G′G′=μ+λG′G+(v-1)G′G2 which depend also on the value of (λ2-4(v-1)μ) and the specific condition on the parameters as follow:When λ=0 this auxiliary equation reduces to Riccati equation, when μ=0 this auxiliary equation reduces to Bernoulli equation and when (λ2≠4(v-1)μ) we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions. Keywords: Two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma, Three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma, Extended simple equation method, Novel G′G-expansion method Traveling wave solutions, Solitary wave solutions

In this research, we investigate one of the most popular model in nature and also industrial which is the pressure equation of bubbly liquids with examination for viscosity and heat transfer which has many application in nature and engineering. Understanding the physical meaning of exact and solitary traveling wave solutions for this equation gives the researchers in this field a great clear vision of the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer and also dynamics of contrast agents in the blood flow at ultrasonic researches. To achieve our goal, we apply three different methods which are extended tanh-function method, extended simple equation method and a new auxiliary equation method on this equation. We obtained exact and solitary traveling wave solutions and we also discuss the similarity and difference between these three method and make a comparison between results that we obtained with another results that obtained with the different researchers using different methods. All of these results and discussion explained the fact that our new auxiliary equation method is considered to be the most general, powerful and the most result-oriented. These kinds of solutions and discussion allow for the understanding of the phenomenon and its intrinsic properties as well as the ease of way of application and its applicability to other phenomena. Keywords: Pressure equation of bubbly liquids with examination for viscosity and heat transfer (the Kudryashov-Sinelshchikov equation), Extended tanh-function method, Extended simple equation method, New auxiliary equation method, Solitary traveling wave solutions, Kink and anti-kink

The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq–Burger and approximate long water wave equations by using the generalized Kudryashov method. The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann–Liouville derivative sense. Applying the generalized Kudryashov method through with symbolic computer maple package, numerous new exact solutions are successfully obtained. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.