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In this paper, we use Paul-Painlev´e approach method, extended rational sine-cosine method and extended rational sinh-cosh method to construct the exact solution of the nonlinear GilsonPickering (GP) equation in plasma. The exact solution of GP equation obtained by the above three methods is new, and we use mathematical software to draw the two-dimensional and three-dimensional graphs of the new exact solutions. Through the study of nonlinear equations in plasma, this study will enrich the research and connotation of nonlinear development equations in plasma. [ABSTRACT FROM AUTHOR]

In this paper, the Gilson-Pickering equation which enables a wave propagation in plasma physics and crystal lattice theory is studied. Based on the auxiliary transformation, the sets of the nonlinear form of ODE along with some analytic solutions are deriven. Multi wave solutions containing one-, two- and three-waves are obtained. The characterizations of new waves, crystal lattice theory and plasma physics using by modified extended tanh-function method are investigated. Through an approach, some solitary solutions are reached. Meanwhile, influence of the coefficients in that equation for the periodic and soliton is illustrated. The effect of the free variables on the behavior of reached plots to a few solutions including exact cases is also explored depending upon the nature of nonlinearities. The dynamic properties of the obtained results are shown and analyzed by some density, two and three-dimensional images. The computational difficulties and outcomes highlight the clarity, effectiveness, and simplicity of the approaches, suggesting that these schemes can be applied to a variety of dynamic and static nonlinear equations governing evolutionary phenomena in computational physics as well as to other real-world situations and a wide range of academic fields. We believe that our results would pave a way for future research generating optical memories based on the nonlinear solitons. [ABSTRACT FROM AUTHOR]

In this study, we investigate the traveling wave solutions of the Gilson-Pickering equation using two different approaches: F-expansion and (1/G')-expansion. To carry out the analysis, we perform a numerical study using the implicit finite difference approach on a uniform mesh and the parabolic-Monge-Ampère (PMA) method on a moving mesh. We examine the truncation error, stability, and convergence of the difference scheme implemented on a fixed mesh. MATLAB software generates accurate representations of the solution based on specified parameter values by creating 3D and 2D graphs. Numerical simulations with the finite difference scheme demonstrate excellent agreement with the analytical solutions, further confirming the validity of our approaches. Convergence analysis confirms the stability and high accuracy of the implemented scheme. Notably, the PMA method performs better in capturing intricate wave interactions and dynamics that are not readily achievable with a fixed mesh. [ABSTRACT FROM AUTHOR]

In this study, we investigate the traveling wave solutions of the Gilson-Pickering equation using two different approaches: F-expansion and (1/G$ ^\prime $)-expansion. To carry out the analysis, we perform a numerical study using the implicit finite difference approach on a uniform mesh and the parabolic-Monge-Ampère (PMA) method on a moving mesh. We examine the truncation error, stability, and convergence of the difference scheme implemented on a fixed mesh. MATLAB software generates accurate representations of the solution based on specified parameter values by creating 3D and 2D graphs. Numerical simulations with the finite difference scheme demonstrate excellent agreement with the analytical solutions, further confirming the validity of our approaches. Convergence analysis confirms the stability and high accuracy of the implemented scheme. Notably, the PMA method performs better in capturing intricate wave interactions and dynamics that are not readily achievable with a fixed mesh.

This paper focuses on obtaining traveling wave solutions of the Fornberg–Whitham model derived from Gilson–Pickering equations, which describe the prorogation of waves in crystal lattice theory and plasma physics by some analytical techniques, i.e., the exp-function method (EFM), the multi-exp function method (MEFM) and the multi hyperbolic tangent method (MHTM). We analyze and compare them to show that MEFM is the optimum method. [ABSTRACT FROM AUTHOR]

This work delves into the analysis of the Gilson–Pickering equation which governs the waves propagation in plasma physics by invoking Lie symmetry analysis. We commence by identifying the Lie point symmetries associated with the equation. These symmetries are then leveraged on to compute the commutator table and subsequently the adjoint representation, ultimately leading to the establishment of an optimal system of one-dimensional subalgebras. Each subalgebra within this system is subsequently utilized to perform symmetry reductions. Through these reductions, various forms of nonlinear ordinary differential equations are obtained, which are subsequently solved using the power series method and Kudryashov’s technique. The resulting solutions are given in terms of hyperbolic functions. To gain deeper insights into the behaviour of these solutions, three-dimensional and two-dimensional plots are presented. Furthermore, applying the Ibragimov’s theorem allows us to derive conserved vectors associated with the Gilson–Pickering equation.

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

In this paper, the Gilson–Pickering (GP) equation with applications for wave propagation in plasma physics and crystal lattice theory is studied. The model with wave propagation in plasma physics and crystal lattice theory is explained. A collection of evolution equations from this model, containing the Fornberg–Whitham, Rosenau–Hyman, and Fuchssteiner–Fokas–Camassa–Holm equations is developed. The descriptions of new waves, crystal lattice theory, and plasma physics by applying the standard tan (ϕ / 2) -expansion technique are investigated. Many alternative responses employing various formulae are achieved; each of these solutions is represented by a distinct plot. Some novel solitary wave solutions of the nonlinear GP equation are constructed utilizing the Paul–Painlevé approach. In addition, several solutions including soliton, bright soliton, and periodic wave solutions are reached using He's variational direct technique (VDT). The superiority of the new mathematical theory over the old one is demonstrated through theorems, and an example of how to design and numerically calibrate a nonlinear model using closed-form solutions is given. In addition, the influence of changes in some important design parameters is analyzed. Our computational solutions exhibit exceptional accuracy and stability, displaying negligible errors. Furthermore, our findings unveil several unprecedented solitary wave solutions of the GP model, underscoring the significance and novelty of our study. Our research establishes a promising foundation for future investigations on incompressible fluids, facilitating the development of more efficient and accurate models for predicting fluid behavior. [ABSTRACT FROM AUTHOR]

This paper presents a finite element scheme for numerical solutions of the Gilson-Pickering (G-P) equation by using septic B-spline functions as approximate functions. Firstly we study optimal-order L²-error estimates for standard Galerkin semi-discrete approximation using smooth splines on a uniform mesh for periodic initial value problem of the G-P equation. A Von-Neumann stability analysis of the algorithm has been performed as well. Moreover, reliableness and practicalness of the presented method is demonstrated by analyzing behavior of single soliton. The L2 and L∞ error norms and two lowest invariants I1 and I2 of the equation have been computed to control proficiency and conservation properties of the suggested algorithm. Obtained numerical results have been illustrated with tables and graphics for easy visualization of properties of the problem modelled. Also the results indicate that our method is favorable. [ABSTRACT FROM AUTHOR]

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences. [ABSTRACT FROM AUTHOR]

The purpose of this article is to achieve new soliton solutions of the Gilson–Pickering equation (GPE) with the assistance of Sardar's subequation method (SSM) and Jacobi elliptic function method (JEFM). The applications of the GPE is wider because we study some valuable and vital equations such as Fornberg–Whitham equation (FWE), Rosenau–Hyman equation (RHE) and Fuchssteiner–Fokas–Camassa–Holm equation (FFCHE) obtained by particular choices of parameters involved in the GPE. Many techniques are available to convert PDEs into ODEs for extracting wave solutions. Most of these techniques are a case of symmetry reduction, known as nonclassical symmetry. In our work, this approach is used to convert a PDE to an ODE and obtain the exact solutions of the NLPDE. The solutions obtained are unique, remarkable, and significant for readers. Mathematica 11 software is used to derive the solutions of the presented model. Moreover, the diagrams of the acquired solutions for distinct values of parameters were demonstrated in two and three dimensions along with contour plots. [ABSTRACT FROM AUTHOR]

In plasma physics and water waves, the Gilson-Pickering equation is an important unidirectional wave propagation model. A large number of analytic wave solutions have been established in the form of hyperbolic, trigonometric, exponential, and rational functions using the advanced auxiliary equation approach in this study. The solutions obtained have been compared with the solutions available in the literature, and it is observed that we have established further wave solutions than the solutions determined by the other methods, namely the (G′/G2)-expansion method, sinh-Gordon approach, etc. This method uses a homogeneous balance rule to estimate a polynomial-type solution and delivers an order. A traveling wave transformation has been used to convert the governing equation into a nonlinear differential equation. Each solution includes a variety of parameters related to the model and method. Using different parameter values, the nature of wave solutions is defined by three-dimensional (3D) and two-dimensional (2D) wave profiles. The solitons change their nature and location for the particular values of the dispersion coefficient α and free parameter κ, and shown in 2D figures to illustrate the different forms of solitons.

In this paper, the modified extended mapping method is applied to obtain solitary wave solutions and other solutions for the Gilson–Pickering equation which describes the wave propagation in plasma physics. Many types of solutions are obtained such as shock wave, singular waves, periodic waves, singular periodic waves, Jacobi elliptic functions, plane waves, hyperbolic waves, Weierstrass elliptic solutions and exponential type solutions. Moreover, 3D graphs of some solutions are introduced for the physical illustration.

In this article, we introduce traveling wave solutions of the Gilson–Pickering equation using two simple methods. Furthermore, we present a numerical study of the Gilson–Pickering equation using the finite difference method. Also, the local truncation error and stability analysis for the difference scheme are presented. We justify the accuracy of the obtained results by exhibiting illustrative tables and proper graphs.

In this work, we developed a hybrid scheme for Gilson–Pickering equation by extracting advantageous features of collocation method and B-splines. Collocation method involves satisfying a differential equation to some tolerance at finite number of points and has low computational cost. The idea is to extract the superior accuracy and smoothness of B-spline functions with the low computational cost of collocation scheme. In the vision of a superficial periodic force, we obtain the quasiperiodic and chaotic behaviors of the forced Gilson–Pickering equation by perusing phase portrait technique, time series and Poincaré section. The sequel of frequency (ω) of the superficial periodic force is very prime in case of transition from weakly chaotic feature to quasiperiodic feature of the forced Gilson–Pickering equation. [ABSTRACT FROM AUTHOR]

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson⁻Pickering model is applied. A set of new complex soliton solutions to the Gilson⁻Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given.

In this paper, a numerical meshless method for solving the Gilson-Pickering equation is considered. The method is based on thin-plate radial basis function using collocation points. The scheme proposed works in a similar fashion as finite-difference method and a predictor-corrector scheme is proposed to avoid solving the nonlinear system. In addition, we use the super not-a-knot method for improving the accuracy at the boundaries. The results of numerical experiments are compared with the existing results in illustrative examples to confirm the accuracy and efficiency of the presented scheme and the norm of the error functions is obtained to show the convergence of the method. [ABSTRACT FROM AUTHOR]

• We established scores of analytical soliton solutions, including the kink-shaped, bell-shaped, U-shaped, parabolic-shaped, periodic solitons, and some other shapes to the Gilson-Pickering equation. • We have shown that the dispersion coefficient has a greater impact than the free parameters in changing the nature and position of solitons. • Soliton profiles have been analysed using 3D and 2D plots of the solutions obtained. In plasma physics and water waves, the Gilson-Pickering equation is an important unidirectional wave propagation model. A large number of analytic wave solutions have been established in the form of hyperbolic, trigonometric, exponential, and rational functions using the advanced auxiliary equation approach in this study. The solutions obtained have been compared with the solutions available in the literature, and it is observed that we have established further wave solutions than the solutions determined by the other methods, namely the (G ′ / G 2)-expansion method, sinh-Gordon approach, etc. This method uses a homogeneous balance rule to estimate a polynomial-type solution and delivers an order. A traveling wave transformation has been used to convert the governing equation into a nonlinear differential equation. Each solution includes a variety of parameters related to the model and method. Using different parameter values, the nature of wave solutions is defined by three-dimensional (3D) and two-dimensional (2D) wave profiles. The solitons change their nature and location for the particular values of the dispersion coefficient α and free parameter κ , and shown in 2D figures to illustrate the different forms of solitons. [ABSTRACT FROM AUTHOR]

In this paper, the modified extended mapping method is applied to obtain solitary wave solutions and other solutions for the Gilson–Pickering equation which describes the wave propagation in plasma physics. Many types of solutions are obtained such as shock wave, singular waves, periodic waves, singular periodic waves, Jacobi elliptic functions, plane waves, hyperbolic waves, Weierstrass elliptic solutions and exponential type solutions. Moreover, 3D graphs of some solutions are introduced for the physical illustration. • Gilson–Pickering equation which describes the wave propagation in plasma physics. • Solitary wave solutions. • Shock wave solutions. • Modified extended mapping method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we show the applicability of the first integral method, which is based on the ring theory of commutative algebra, to the regularized long-wave Burgers equation and the Gilson–Pickering equation under a parameter condition. Our method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are derived in a concise manner. [Copyright &y& Elsevier]

Abstract: In this paper, the qualitative behavior and exact travelling wave solutions of the Gilson–Pickering equation are studied by using the qualitative theory of polynomial differential system. The phase portraits of the system are given under different parametric conditions. Some exact travelling wave solutions of the Gilson–Pickering equation are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given. [Copyright &y& Elsevier]

In this article, we introduce traveling wave solutions of the Gilson–Pickering equation using two simple methods. Furthermore, we present a numerical study of the Gilson–Pickering equation using the finite difference method. Also, the local truncation error and stability analysis for the difference scheme are presented. We justify the accuracy of the obtained results by exhibiting illustrative tables and proper graphs. • We introduce traveling wave solutions of the Gilson–Pickering equation using two simple methods. • Furthermore, we present a numerical study of the Gilson–Pickering equation using the finite difference method. • Also, the local truncation error and stability analysis for the difference scheme are presented. • We justify the accuracy of the obtained results by exhibiting illustrative tables and proper graphs. [ABSTRACT FROM AUTHOR]

In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given. [ABSTRACT FROM AUTHOR]