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In this paper, the optical soliton and solitary wave solutions of the (2 + 1) -dimensional Mel'nikov equation are investigated using the Kudryashov R function technique. The Kudryashov R function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the (2 + 1) -dimensional Mel'nikov equation have been displayed in 3D, 2D and contour graphs. [ABSTRACT FROM AUTHOR]

In this paper, a reliable and effective local discontinuous Galerkin (LDG) scheme for numerically solving the classical Rosenau–Hyman equation with non-periodic boundary conditions has been proposed. This study employs the third-order nonlinearly stable total variation diminishing Runge–Kutta method and the LDG method, respectively, to discretize the temporal and spatial derivatives. Finally, numerical simulations are performed on various test problems and compared with the exact results as well as results produced by a few other numerical methods, to analyze the reliability and efficiency of the proposed method. The results generated, which validate the expected order of accuracy, are presented through multiple tables. In addition, several graphical representations of the problem are presented to depict the behavior of the solution. [ABSTRACT FROM AUTHOR]

This paper investigates the (3 + 1) -dimensional extended Sakovich equation, which represents an essential nonlinear scientific model in the field of ocean physics. The Lie symmetry analysis has been utilized for extracting the non-traveling wave solutions of the (3 + 1) -dimensional extended Sakovich equation. These solutions are investigated through infinitesimal generators, which are obtained from Lie's continuous group of transformations. As there are infinite possibilities for the linear combination of infinitesimal generators, so a one-dimensional optimal system of subalgebra has been established using Olver's standard approach. Moreover, by considering the optimal system of subalgebra, the extended Sakovich equation is converted into a solvable nonlinear PDE through symmetry reductions. Finally, the conservation laws for the governing equation have been derived using Ibragimov's generalized theorem and quasi-self-adjointness condition. [ABSTRACT FROM AUTHOR]

The (2 + 1) -dimensional generalized Hirota–Satsuma–Ito equation describing the numerous wave dynamics in shallow waters is investigated in this study. The integrable characteristics of the aforesaid equation, such as a bilinear Bäcklund transformation and Lax pair, are revealed using the Bell polynomials method. First, using this technique, with the aid of Hirota operators, the bilinear form is constructed for the considered equation. In addition, the bilinear Bäcklund transformation and the Lax pair of the aforesaid equation are derived successfully using the bilinear form. Moreover, the bilinear form is also used to construct analytical solutions utilizing the three-wave approach with a test function. While using this method, numerous analytical solutions are derived, which are not presented in the literature. A three-dimensional graph has been plotted for each of the obtained results by giving the appropriate values of the free parameters. These plots reveal a wide variety of wave behavior, such as kink-soliton, periodic wave, anti-kink soliton, and complex periodic wave solutions. [ABSTRACT FROM AUTHOR]

The time-dependent variable coefficients of Bogoyavlensky–Konopelchenko (BK) equation and generalized Bogoyavlensky–Konopelchenko (gBK) equation are considered in this paper. The integrability test by Painlevé analysis is being implemented on both the considered equations. An auto-Bäcklund transformation has been generated with the help of Painlevé analysis for both equations. Auto-Bäcklund transformation method has been used for obtaining the analytic solutions. By using auto-Bäcklund transformation method, three different analytic solution families have been derived for each of the considered equations. Multi-soliton solutions are also calculated for both the considered equations by using Hereman and Nuseir algorithm. All the results are expressed graphically in 3D by varying different functions and parametric values. These graphs reveal the physical significance of equations under consideration. [ABSTRACT FROM AUTHOR]

In this paper, the local discontinuous Galerkin method is used to analyze numerical solutions for nonlinear Allen–Cahn equations with nonperiodic boundary conditions. To begin with, the spatial variables are discretized to generate a semidiscrete method of lines scheme. This yields an ordinary differential equation system in the temporal variable, which is then solved using the higher-order total variation diminishing Runge–Kutta method. A comparison of the generated numerical results to the exact results for various test problems using different tables and figures provides insight into the effectiveness and accuracy of the proposed method. The numerical results confirm that the proposed method is an effective numerical scheme for solving the Allen–Cahn equation since the obtained solutions are extremely close to the exact solutions while exhibiting substantially less error. [ABSTRACT FROM AUTHOR]

This paper considers the generalized KdV6 equation with time-dependent variable coefficients. The integrability of the considered equation is being examined by the Painlevé analysis method. Further, an auto-Bäcklund transformation method has been adopted to obtain the analytic solutions. Using this technique, five novel families of analytic solutions in the form of rational, exponential, hyperbolic and trigonometric functions have been successfully found for the considered equation. New kink-antikink and periodic soliton solutions have been discovered using this method. The solutions are graphically depicted to show the physical significance of the problem under consideration. [ABSTRACT FROM AUTHOR]

The governing equations for fluid flows, i.e. Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) model equations represent a water wave model. These model equations describe the bidirectional propagating water wave surface. In this paper, an auto-Bäcklund transformation is being generated by utilizing truncated Painlevé expansion method for the considered equation. This paper determines the new bright soliton solutions for (2 + 1) and (3 + 1) -dimensional nonlinear KP-BBM equations. The simplified version of Hirota's technique is utilized to infer new bright soliton solutions. The results are plotted graphically to understand the physical behavior of solutions. [ABSTRACT FROM AUTHOR]

In this paper, a novel meshless numerical scheme to solve the time-fractional Oskolkov–Benjamin–Bona–Mahony–Burgers-type equation has been proposed. The proposed numerical scheme is based on finite difference and Kansa-radial basis function collocation approach. First, the finite difference scheme has been employed to discretize the time-fractional derivative and subsequently, the Kansa method is utilized to discretize the spatial derivatives. The stability and convergence analysis of the time-discretized numerical scheme are also elucidated in this paper. Moreover, the Kudryashov method has been utilized to acquire the soliton solutions for comparison with the numerical results. Finally, numerical simulations are performed to confirm the applicability and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

The mathematical modeling of physical systems is generally governed by evolution equations in nonlinear form. Therefore, it is critical to obtain exact analytical solutions to these equations. This paper contains the application of improved (G ′ /G)-expansion and extended (G ′ /G)-expansion methods for establishing new analytical solutions of a nonlinear evolution equation. Mainly, improved (G ′ /G)-expansion and extended (G ′ /G)-expansion methods have been applied to time-fractional Kaup-Kupershmidt equation for getting some new general forms of exact solutions. Here, algorithms provide a more convenient and systematically handling the solution process of the non-linear evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, new traveling wave solutions of time fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation have been revealed by the help of extended Kudryashov's method. In this case, the time fractional derivative has been considered in the sense of conformable fractional derivative. The salient feature of the proposed extended Kudryashov's method is to take full advantage of solutions for the Riccati and the Bernoulli equations involving parameters. Using some particular wave transformation, the given equation is converted to the nonlinear ODE by the help of chain rule and the derivative of a composite function; and then, a novel analytical method viz. extended Kudryashov's method has been used to solve the resulting equation. Consequently, some new exact solutions have been successfully obtained. The exact solutions devised by the proposed method manifest that the proposed method is effective and easy to implement. [ABSTRACT FROM AUTHOR]

A comparison between two numerical approximation methods i.e., Euler-Murayama and 1.5 strong Taylor methods have been established in this article. The stochastic point reactor kinetics equations consist of a system of Itô stochastic differential equations (SDEs) and this system is solved over each time-step size using Euler-Murayama and 1.5 strong Taylor methods. The obtained results establish the accuracy of both the methods in solving the stochastic point reactor kinetics equations in presence of Newtonian temperature feedback. [ABSTRACT FROM AUTHOR]

In this paper, new types of Jacobi elliptic function solutions of Dodd–Bullough–Mikhailov(DBM) and Tzitzeica–Dodd–Bullough(TDB) equations have been obtained using a new extended auxiliary equation method. A new family of explicit travelling wave solutions is derived. The solitary wave solutions and periodic solutions for these equations are formally derived from the Jacobi elliptic function solutions. The proposed method has been efficiently applied to solve the DBM and TDB equations. [ABSTRACT FROM AUTHOR]

The primary goal of this research is to carry out the closed-form analytical solutions of the generalised (1+1)-dimensional Kersten-Krasil'shchik coupled KdV-mKdV equations, which are frequently used to represent shallow water waves, elastic media, electric circuits, electrodynamics, etc. By following the Lie symmetry analysis for this coupled system, optimal system of subalgebras of the secured symmetries has been constructed. Furthermore, the given system is reduced to the system of ordinary differential equations by utilizing an optimal system of subalgebra. Using power series solutions and their convergence analysis, explicit exact solutions of the reduced ordinary differential equations were obtained. Additionally, the conservation laws of the aforementioned coupled system are derived by applying the "new conservation theorem" proposed by Ibragimov. [ABSTRACT FROM AUTHOR]