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This investigation employs contemporary computational and numerical techniques to derive analytical and approximate soliton solutions for the Caudrey–Dodd–Gibbon model, which represents a significant variation of the fifth-order Korteweg–de Vries equation. Diverse analytical solutions are constructed, employing distinct formats such as exponential, trigonometric, and hyperbolic functions. Simulations, including two-dimensional, three-dimensional, contour, polar, and discrete plots, are presented to illustrate the real-world behavior of a single soliton. Furthermore, these solutions are utilized to evaluate the essential conditions for implementing the proposed numerical scheme. The agreement between the computed and approximate solutions is demonstrated through various techniques. These results unequivocally establish the superiority of these methods for solving nonlinear mathematical physics problems. [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]

This study uses crystal lattice theory and physicochemical characterization to show a number of correct wave solutions that are like the way plasma waves move. The nonlinear time–fractional Gilson–Pickering (G P) model has been addressed using two distinct analytical and numerical techniques. This model is used in crystal lattice theory and plasma physics to show how waves move, so it is a basic model for how waves move in one direction. Utilizing the modified rational and He's variational iteration approximations in conjunction with the β–fractional derivative principles, we give the handled model fresh and precise solitary wave solutions. Various contour, three-dimensional, and two-dimensional graphs depict the determined outcomes. Compared to other recent studies, ours indicates the importance of this research area. The presented methods show how simple, direct, and effective they are and how they can be used with a wide range of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This study presents fundamental theorems, lemmas, and mapping definitions. There are three types of mappings: binary operators, compatible mappings, and sequentially continuous mappings. The symbols used to represent fuzzy metric spaces are intuitive. Icons were also used to prescribe a shared, linked fixed point in intuitionistic fuzzy metric space for two compatible and sequentially continuous mappings that satisfy ϕ-contractive conditions. To accomplish this, finding the intersection of both mappings was necessary. [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]

This study investigates the physical significance of the dimensionless Zakharov equation ZE , which theoretically describes the interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves. Coastal engineering, fluid dynamics, nonlinear optics, and plasma wave physics are all fields of study that explore this relationship. We use the modified Khater method, the auxiliary equation approach, and the Khater II method, all cutting-edge ways to compute, to come up with a wide range of new and different solutions. Other physical aspects of this interaction are briefly explained below. To figure out the originality of our solutions, we have to compare our results to those of other published studies. [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, the nonlinear fractional Lotka–Volterra model is analyzed and numerically studied. This research is based on applying the three latest analytical schemes and three other numerical schemes to construct rich wave solutions. In different forms, many novel solitary wave solutions are built and presented in two-dimensional, three-dimensional, and contour plots. The numerical method conditions are evaluated through the obtained analytical solutions, and the accuracy of the analytical solutions is studied. Many numerical solutions are constructed based on the employed schemes. Additionally, the analytical, semi-analytical, numerical, and absolute values of error between the values of obtained solutions are calculated with the different values of the given variable in the solutions. Furthermore, the match between the obtained analytical solution and the numerical solution has been explained through some two-dimensional distributed radar charts. The contribution of this article is demonstrated by comparing the obtained solution with the recently published results of the same model. [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]

The analytical and semi-analytical solutions to the quadratic–cubic fractional nonlinear Schrödinger equation are discussed in this research article. The model's fractional formula is transformed into an integer-order model by using a new fractional operator. The theoretical and computational approaches can now be applied to fractional models, thanks to this transition. The application of two separate computing schemes yields a large number of novel analytical strategies. The obtained solutions secure the original and boundary conditions, which are used to create semi-analytical solutions using the Adomian decomposition process, which is often used to verify the precision of the two computational methods. All the solutions obtained are used to describe the shifts in a physical structure over time in cases where the quantum effect is present, such as wave-particle duality. The precision of all analytical results is tested by re-entering them into the initial model using Mathematica software 12. [ABSTRACT FROM AUTHOR]

This paper investigates the analytical solutions of the perturbed nonlinear Schrödinger equation through the modified Khater method. This method is considered one of the most recent accurate analytical schemes in nonlinear evolution equations where it obtained many distinct forms of solutions of the considered model. The investigated model in this paper is an icon in quantum fields where it describes the wave function or state function of a quantum-mechanical system. The physical characterization of some obtained solutions in our study is explained through sketching them in two- and three-dimensional contour plots. The novelty of our study is clear by showing the matching between our solutions and those that have been constructed in previously published papers. [ABSTRACT FROM AUTHOR]

This article studies the generalized (2 + 1)-dimensional shallow water equation by applying two recent analytical schemes (the extended simplest equation method and the modified Kudryashov method) for constructing abundant novel solitary wave solutions. These solutions describe the bidirectional propagating water wave surface. Some obtained solutions are sketched in two- and three-dimensional and contour plots for demonstrating the dynamical behavior of these waves along shallow water. The accuracy of the obtained solutions and employed analytical schemes is investigated using the evaluated solutions to calculate the initial condition, and then the well-known variational iterational (VI) method is applied. The VI method is one of the most accurate semi-analytical solutions, and it can be applied for high derivative order. The used schemes' performance shows their effectiveness and power and their ability to handle many nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this article, two recent computational schemes [the modified Khater method and the generalized exp − φ (I) –expansion method] are applied to the nonlinear predator–prey system for constructing novel explicit solutions that describe a prototype of an excitable system. Many distinct types of solutions are obtained such as hyperbolic, parabolic, and rational. Moreover, the Hamiltonian system's characteristics are employed to check the stability of the obtained solutions to show their ability to be applied in various applications. 2D, 3D, and contour plots are sketched to illustrate more physical and dynamical properties of the obtained solutions. Comparing our obtained solutions and that obtained in previous published research papers shows the novelty of our paper. The performance of the two used analytical schemes explains their effectiveness, powerfulness, practicality, and usefulness. In addition, their ability in employing various forms of nonlinear evolution equations is also shown. [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 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]

In this paper, we examine a modified auxiliary equation method. We applied this novel method on Wu-Zhang system. This model used to describe (1 + 1)-dimensional dispersive long wave in two horizontal directions on shallow waters. This model is one of the fractional nonlinear partial differential equations. We used conformable derivatives properties to convert nonlinear fractional partial differential equation into the ordinary differential equation with integer order. We obtained many different kinds of solutions such as kink and anti-kink, dark, bright, shock, singular, periodic solitary wave. [ABSTRACT FROM AUTHOR]