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This study comprehensively explores the (3 + 1) -dimensional Mikhailov–Novikov–Wang (ℕ) integrable equation, with the primary objective of elucidating its physical manifestations and establishing connections with analogous nonlinear evolution equations. The investigated model holds significant physical meaning across various disciplines within mathematical physics. Primarily, it serves as a fundamental model for understanding nonlinear wave propagation phenomena, offering insights into wave behaviors in complex media. Moreover, its relevance extends to nonlinear optics, where it governs the dynamics of optical pulses and solitons crucial for optical communication and signal processing technologies. Employing analytical methodologies, namely the unified () , Khater II ( hat.II) method, and He's variational iteration (ℍ ) method, both numerical and analytical solutions are meticulously examined. Through this investigation, the intricate behaviors of the equation are systematically unveiled, shedding illuminating insights on various physical phenomena, notably including wave propagation in complex media and nonlinear optics. The outcomes not only underscore the efficacy of the analytical techniques deployed but also accentuate the equation's pivotal role in modeling a broad spectrum of nonlinear wave dynamics. Consequently, this research significantly advances our comprehension of complex physical systems governed by nonlinear dynamics, thereby contributing notably to interdisciplinary pursuits in mathematical physics. [ABSTRACT FROM AUTHOR]

This study introduces innovative analytical solutions for the (2 + 1) -dimensional nonlinear Jaulent–Miodek () equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of waves. Within the realm of nonlinear fluid dynamics, the equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the (2 + 1) -dimensional equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field. [ABSTRACT FROM AUTHOR]

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 aims to solve the nonlinear Pochhammer–Chree (ℕℙℂ) equation to understand its physical implications and establish connections with other nonlinear evolution equations, particularly in plasma dynamics. By using the Khater II (핂hat. II) method for analytical solutions and validating these solutions numerically with the variational iteration method, this study offers a detailed understanding of the equation’s behavior. The results demonstrate the effectiveness of these methods in accurately modeling the system, highlighting the importance of combining analytical and numerical approaches for reliable solutions. This research significantly advances the field of nonlinear dynamics, especially in plasma physics, by employing multidisciplinary methods to tackle complex physical processes. Moreover, the ℕℙℂ equation is relevant in various physical contexts beyond plasma dynamics such as optical and quantum fields. In optics, it models the propagation of nonlinear waves in fiber optics, where similar nonlinear evolution equations describe wave interactions. In quantum field theory, the ℕℙℂ equation helps in understanding the behavior of quantum particles and fields under nonlinear effects, making it a versatile tool for studying complex phenomena across different domains of physics. [ABSTRACT FROM AUTHOR]

In this study, we investigate the complex dynamics of the (1+1)-dimensional generalized Burgers–Fisher (GBF) model, a nonlinear partial differential equation that encapsulates the interplay between wave propagation, diffusion, and reaction processes. Our work employs a combination of the modified Khater (MKhat) method, the unified (UF) method, and He’s variational iteration (HVI) scheme to derive and validate analytical and numerical solutions. We present a comprehensive analysis of solitary wave, shock wave, and diffusion-driven phenomena within the GBF framework. The novelty of our study lies in the integration of these methods to provide deeper insights into the model’s physical implications, specifically highlighting the interactions between nonlinear advection, diffusion, and reaction mechanisms. This approach not only enhances the accuracy and applicability of the derived solutions, but also contributes to the advancement of nonlinear wave theory and related interdisciplinary fields. [ABSTRACT FROM AUTHOR]

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 investigation aims to examine the resolution of the generalized nonlinear time fractional Harry Dym (ℕ ℍ) equation through a combined application of analytical and numerical methodologies. The primary objective is to scrutinize the equation's behavior and present proficient methodologies for its resolution. The Khater II analytical technique and numerical frameworks, specifically the Cubic-B-spline, Quantic-B-spline, and Septic-B-spline schemes, are proposed for this purpose. The outcomes of this inquiry yield substantial revelations regarding the characteristics of ℕ ℍ equation. Both the analytical approach and numerical schemes demonstrate their efficacy in yielding precise solutions. These findings carry noteworthy implications across various disciplines, encompassing physics, mathematics, and engineering. The investigated model appears in plasma physics research on soliton theory and nonlinear waves. Soliton waves, which keep their shape and velocity while propagating, are found in plasma physics and other domains. In plasma environments, the Harry Dym equation describes these solitons and their behavior. Solitons are essential for understanding plasma dynamics, including nonlinear waves and structures. They help comprehend plasma dynamics, wave interactions, and other nonlinear processes. The principal deductions drawn from this study underscore the effectiveness and viability of the proposed techniques in resolving the ℕ ℍ equation. This research introduces innovative contributions in terms of insights and methodologies pertinent to analogous nonlinear fractional equations. Its scope encompasses nonlinear dynamics, fractional calculus, and numerical analysis. [ABSTRACT FROM AUTHOR]

This study investigates the nonlinear time-fractional Harry Dym ( ℍ ) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym (ℍ ) equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative introduces a memory effect, allowing the model to capture nonlocal interactions and long-range dependencies in the wave dynamics. The primary objective of this research is to obtain accurate analytical solutions to the ℍ equation and explore its physical characteristics. We employ the Khater III method as the primary analytical technique and utilize the He's variational iteration (ℍ ) method as a numerical scheme to validate the obtained solutions. The close agreement between analytical and numerical results enhances the applicability of the solutions in practical applications of the model. This research contributes to a deeper understanding of the ℍ equation's behavior, particularly in the presence of fractional-order dynamics. The obtained solutions provide valuable insights into the complex interplay between nonlinearity and memory effects in the wave propagation phenomena described by the model. By shedding light on the physical characteristics of the ℍ equation, this study paves the way for further investigations into its potential applications in diverse physical settings. [ABSTRACT FROM AUTHOR]

This paper proposes abundant accurate wave solutions that represent the plasma wave propagation based on crystal lattice theory and physicochemical characterization. The Gilson–Pickering () model is analytically and numerically solved by two recent techniques. This model is a basic unidirectional wave propagation model that describes the prorogation of waves in crystal lattice theory and plasma physics. The investigated model has a deep connection with some nonlinear evolution equations under specific values of its parameters, such as the Fuchssteiner–Fokas–Camassa–Holm (ℱ ℱ ℋ) equation, the Rosenau–Hyman (ℛ ℋ) equation, and the Fornberg–Whitham (ℱ ) equation. The Sardar sub-equation and He's variational iteration techniques are employed to construct novel and accurate solitary wave solutions of the handled model. The obtained solutions are explained through some distinct graphs in contour, three-, two-dimensional. The research value is explained by comparing our results with some recently published studies. The employed methods show their simple, direct, effectiveness and their ability for handling many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

Understanding the intricate Korteweg–de Vries ( c K d V ) equation is paramount for comprehending various nonlinear wave phenomena, owing to its capacity to depict the propagation of nonlinear waves in media characterized by complex-valued dispersive and nonlinear coefficients. The practical ramifications of the c K d V equation are extensive, spanning disciplines such as optics, plasma physics, and other realms dealing with intricate media. This investigation employs the Khater II method and a generalized rational approach as analytical methodologies to generate novel exact solutions for the c K d V equation. The Khater II method is employed to transform the nonlinear partial differential equation into a nonlinear ordinary differential equation, thus facilitating a systematic resolution. Moreover, the generalized rational technique utilizes a rational ansatz function to derive diverse forms of solutions. Application of these methodologies leads to the discovery of fresh exact solutions, encompassing solitary and periodic wave solutions for the c K d V equation. These solutions are expressed in rational forms featuring arbitrary functions, thereby expanding the repertoire of known solutions for the model. The efficacy of the analytical methodologies employed becomes evident through the discovery of these novel exact solutions, thereby enriching our understanding of the physical interpretations and wave characteristics associated with the c K d V equation. The derived solutions augment the existing body of knowledge pertaining to the model. This research enhances our comprehension of the c K d V equation, thereby advancing nonlinear wave analysis with potential applications in physics, optics, plasma science, and allied engineering domains. Future investigations may explore the extension of these methodologies to address other nonlinear wave equations. [ABSTRACT FROM AUTHOR]

This study investigates the integrableMikhailov-Novikov-Wang (MNW) equation in (1+1)- dimensional space-time, utilizing rigorous analytical methodologies including the Unified (UF) and Khater II (Khat.II) methods, alongside numerical solutions. Situated within the established mathematical framework of physics, the MNW equation holds significant relevance in elucidating various physical phenomena, encompassing solitons, nonlinear wave propagation, quantum mechanics, and field theories, contingent upon the specific context and parameters selected for analysis. The primary objective of this study is to conduct a comprehensive analysis of the MNW equation and propose effective resolution techniques. This is achieved through the amalgamation of diverse analytical and numerical methodologies, facilitating an in-depth exploration of the equation's characteristics. Noteworthy findings indicate that the UF, Khat.II methods, in conjunction with He's variational iteration method, yield robust and highly accurate solutions. The significance of these results lies in their potential to address complex nonlinear equations, providing researchers and practitioners with versatile tools for analysis and interpretation. This research contributes a novel perspective by skillfully integrating these analytical methodologies, thereby advancing the field of mathematical physics and nonlinear differential equations. It is imperative to note that this study is purely theoretical in nature and does not involve specific subjects or participants. The approach adopted is grounded in meticulous numerical experimentation and analytical scrutiny, ensuring rigor and reliability in the findings presented. [ABSTRACT FROM AUTHOR]

This study presents a comprehensive investigation into the soliton solutions for the Khokhlov–Zabolotskaya–Kuznetsov ( ) problem using the Bernoulli sub-equation function approach, generalized Kudryashov method, and Homotopy perturbation method. The equation, a nonlinear wave equation that governs the propagation of high-amplitude ultrasonic waves through nonlinear media, is studied in detail. Perturbation theory and asymptotic expansions are employed to derive approximate solutions for the problem. These methodologies offer valuable insights into the behavior of ultrasonic waves in different media, thereby facilitating the optimization of ultrasonic transducer design and enhancing the precision of ultrasonic imaging systems. The analytical and semi-analytical methods utilized for solving the equation are computationally efficient and serve as valuable resources for researchers and engineers working in the field of ultrasonics. The outcomes of this study carry significant implications for the understanding of ultrasonic wave behavior in nonlinear and dissipative media, ultimately contributing to the development of more accurate and efficient ultrasonic imaging techniques. [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]

This study employs three advanced computational and numerical techniques to solve the nonlinear fractional modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV–ZK) equation in magnetized plasma. The reductive perturbation approach is utilized to investigate the dynamics of various components, namely isothermal species, immobile background species, and warm adiabatic fluid, in magnetized plasma. Emphasis is placed on unraveling the asymmetrical propagation characteristics of nonlinear electrostatic waves. The model's solutions encompass diverse types of solitons, including ion-acoustic, dust acoustic, and electron acoustic solitons. Analytical solutions are obtained using a variety of mathematical functions, such as exponents, trigonometry, and hyperbolas. Two- and three-dimensional density graphs illustrate the practical behavior of a single soliton. The primary objective of employing numerical schemes is to assess the accuracy of the derived solutions, and the outcomes demonstrate the efficacy of the analytical method in solving nonlinear mathematical and physical problems. Several techniques are employed to validate the consistency between calculated and estimated results, ensuring the study's accuracy and reliability. Overall, this investigation underscores the effectiveness of numerical and analytical techniques in tackling complex mathematical models, offering a promising avenue for future research in the field. The findings carry significant implications for comprehending nonlinear phenomena in magnetized plasma and contribute to advancing the field. [ABSTRACT FROM AUTHOR]

Approximation and analysis are used for investigating accurate soliton solutions of the ill-posed Boussinesq (IPB) equation. The investigated model explains shallow-water gravitational waves. It examines one-dimensional nonlinear strings and lattices. IPB explains small-amplitude surface waves on nonlinear strings and lattices. We provide unique analytical solutions to analyze numerical beginning and boundary conditions. A solution's quality is judged by its divergence from analytical predictions. Physical wave properties are illustrated. [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]

In this research, analytical and semi-analytical soliton solutions for the nonlinear fractional (2 + 1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation (F C B S E) in the non-local form are obtained using recent computational and numerical methods. The F C B S E is a significant model for investigating various phenomena, such as internal ocean waves, tsunamis, river tidal waves, and magneto-sound waves in plasma. The constructed solution helps in understanding the interaction between a long wave moving along the x-axis and a Riemann wave propagating along the y-axis. Various analytical solutions, such as exponential, trigonometric, and hyperbolic, have been formulated differently for this model, which is a specific derivation of the well-known Korteweg–de Vries equation. Density charts in two and three dimensions are used to visualize the behavior of a single soliton in reality through simulations. The results demonstrate the effectiveness of the employed numerical scheme and various methods to ensure the consistency of computational and approximation answers. Overall, this study demonstrates the potential of recent computational and numerical techniques for solving nonlinear mathematical and physical problems. [ABSTRACT FROM AUTHOR]

In partial differential equations, the generalized modified equal-width (G M E W) equation is commonly used to model one-dimensional wave propagation in nonlinear media with dispersion processes. In this article, we use two modern, accurate analytical and numerical techniques to find the exact traveling wave solutions for the model we are looking at. The results are new, and at present, they can be used in many different areas of research, such as engineering and physics. The proposed numerical method is helpful because it gives an estimate on the accuracy of the solutions. Distinct graphs, such as a contour plot, a two-dimensional graph, and a three-dimensional graph, were used to show the analytical and numerical results. Using symbolic computation, we demonstrate that our approach is a powerful mathematical tool that can be applied to a wide range of nonlinear wave problems. [ABSTRACT FROM AUTHOR]

The direct influence of the integrability requirement on mixed derivative nonlinear Schrödinger equations is investigated in this paper. A. Rangwala mathematically formalized these effects in 1990 and dubbed this form the Rangwala–Rao ( R R ) equation. Our research focuses on innovative soliton wave solutions and their interactions in order to provide a clear picture of the slowly evolving envelope of the electric field and pulse propagation in optical fibers in terms of the dispersion effect. For creating unique solitary wave solutions to the investigated model, three contemporary computational strategies (extended direct (ExD) method, improved F–expansion (ImFE) method, and modified Kudryashov (MKud) method) are employed. These solutions are numerically computed to demonstrate the dynamical behavior of optical fiber pulse propagation. The originality of the paper's findings is proved by comparing our results to previously published results. [ABSTRACT FROM AUTHOR]

This paper investigates the soliton wave solution of the nonlinear biological population (NBP) model by employing a novel computational scheme. The selected model for this study describes the logistics of the population because of births and deaths. Some novel structures of the NBP model's solutions, are obtained such as exponential, trigonometric, and hyperbolic. These solutions are clarified through some distinct graphs in contour three plot, three-dimensional, and two-dimensional plots. The Hamiltonian system's characterizations are used to check the obtained solutions' stability. The solutions' accuracy is checked by handling the NBP model through the variational iteration (VI) method. The matching between analytical and semi-analytical solutions shows the accuracy of the obtained solutions. The method's performance shows its effectiveness, power, and ability to apply to many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This paper studies the well-known mathematical model's analytical wave solutions (modified Benjamin–Bona–Mahony (BBM) equation), which demonstrates the propagation of long waves in the nonlinear dispersive media in a visual illusion. Six recent analytical and semi-analytical schemes (extended simplest equation (ESE) method, modified Kudryashov (MKud) method, sech–tanh expansion method, Adomian decomposition (ADD) method, El Kalla (EK) expansion method, variational iteration (VI) method) are applied to the considered model for constructing abundant analytical and semi-analytical novel solutions. This variety of solutions aims to investigate the analytical techniques' accuracy by calculating the absolute error between analytical and semi-analytical solutions that shows the matching between them. The analytical results are sketched through two-dimensional (2D), three-dimensional (3D), contour plot, spherical plot and polar plot. The stability characterization of the analytical solutions is investigated through the Hamiltonian system's features. The originality and novelty of this paper are discussed, along with previously published papers. [ABSTRACT FROM AUTHOR]

This paper explores accurate, stable, and novel soliton wave solutions of the nonlinear Qiao model. This model, which was derived in 2007, possesses a Lax representation and bi-Hamiltonian structure, where it is a second positive member in the utterly integrable hierarchy. The well-known generalized extended tanh-function method is employed to construct novel soliton wave solutions. The stable property of the obtained solutions is examined along with the Hamiltonian system's characterizations. Furthermore, the accuracy of the obtained solutions is checked by comparing it with the model's semi-analytical solutions that have been obtained by employing the variational iteration (VI) method. The obtained analytical and semi-analytical solutions are demonstrated through some distinct graphs to show more physical and dynamical behavior of the investigated model. The used analytical and semi-analytical schemes' performance is checked to show if it is effective and powerful. [ABSTRACT FROM AUTHOR]

This study performs two recent methods on the nonlinear long-short wave interaction system (NLSWIS) to obtain novel formulas of solitary solutions representing an optical field that does not alter by multiplication a sensitive balance between the different terms impacts in the medium. Elastic medium is defined as the medium that can adjust in the figure due to a deforming strength and come back shape to its original form when the force eliminates. The wave is produced by vibrations that are a consequence of acoustic power. Also known as the sound wave, or acoustic wave. For examples of elastic mediums are air, water and so on. The improved G ′ G expansion method and novel G ′ G expansion method are implemented, which are considered as two general methods in this field. Applying these methods to many nonlinear partial differential equations gives diverse and different forms and configuration solutions that allow users of those two methods a large number of solutions that provide broader scope to use. [ABSTRACT FROM AUTHOR]

This document is a retraction notice from AIP Advances regarding four manuscripts that have been retracted due to evidence of manipulation in the publication and peer review processes. The editors of AIP Advances have retracted these papers after questions were raised about the data processing and analysis. The retraction is based on the discovery of identical reviewer reports and requests for revisions to add citations to unrelated works by the authors, indicating peer review manipulation. The authors maintain that the data support the validity of the papers, but the editors believe that the issues with the peer review process undermine their confidence in the validity of the papers. [Extracted from the article]

This document is a retraction notice from AIP Advances regarding four manuscripts that have been retracted due to evidence of manipulation in the publication and peer review processes. The editors of AIP Advances have determined that critical steps in the peer review process were manipulated, including the return of identical reviewer reports and author-suggested reviewers requesting revisions to add citations to unrelated works by the authors. As a result, the validity of these papers has been called into question and they are considered potentially unreliable. The authors maintain that the data support the validity of the papers, but the editors believe that the issues with the peer review process undermine confidence in their validity. [Extracted from the article]

This document is a retraction notice from AIP Advances regarding a manuscript titled "Exploring the wave solutions of a nonlinear non-local fractional model for ocean waves." The manuscript, along with three others, has been retracted due to evidence of manipulation in the publication and peer review processes. AIP Publishing requests that readers consider these works potentially unreliable as they have not undergone a credible peer-review process. The editors of AIP Advances have retracted the papers after questions were raised about the data processing and analysis. The authors maintain that the data support the validity of the papers, but the issues with the peer review process have undermined confidence in their validity. [Extracted from the article]

The accuracy of novel lump solutions of the potential form of the three–dimensional potential Yu–Toda–Sasa–Fukuyama (3-Dp-YTSF) equation is investigated. These solutions are obtained by employing the extended simplest equation (ESE) and modified Kudryashov (MKud) schemes to explore its lump and breather wave solutions that characterizes the dynamics of solitons and nonlinear waves in weakly dispersive media, plasma physics, and fluid dynamics. The accuracy of the obtained analytical solutions is investigated through the perspective of numerical and semi-analytical strategies (septic B-spline (SBS) and variational iteration (VI) techniques). Additionally, matching the analytical and numerical solutions is represented along with some distinct types of sketches. The superiority of the MKud is showed as the fourth research paper in our series that has been beginning by Mostafa M. A. Khater and Carlo Cattani with the title "Accuracy of computational schemes". The functioning of employed schemes appears their effectual and ability to apply to different nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, the weakly nonlinear shallow-water wave model is mathematically investigated by applying the modified Riccati-expansion method and Adomian decomposition method. This model is used to describe the propagation of waves in weakly nonlinear and dispersive media. We obtain exact and solitary wave solutions of this model by using the modified Riccati-expansion method then using these solutions to determine the boundary and initial conditions. These conditions are employed to evaluate the semi-analytical wave solutions and calculate the absolute value of error. The values of absolute error show the accuracy of the obtained solutions. Some solutions are sketched to show the perspective view of the solution of this model. Moreover, the novelty of the obtained solutions is illustrated by showing the similarity and differences between our and previous solutions of the model. [ABSTRACT FROM AUTHOR]