David Yaro

In the present research, we explored the various kinds of optical solitons and many other solitary wave solutions for the nonlinear Akbota equation by utilizing the symbolic computational simulation on the basis of the improved F-expansion approach. The nonlinear Akbota equation has applications in physics and engineering. The examined solitary wave and soliton solutions have interesting physical structures, including anti-kink wave solitons, bright solitons, kink wave solitons, dark solitons, periodic wave solitons, peakon bright solitons, peakon dark solitons, mixed bright–dark periodic solitons, mixed solitons in bright–dark form, and solitary wave structures. The newly extracted soliton solutions in this study shed light on the fact that the utilized approach is more efficient, concise, powerful, effective, straightforward, and simple, and we can also utilize it for other higher order nonlinear complex models. The extracted solutions will be helpful to understand the nonlinear phenomena in various areas of nonlinear sciences and engineering, including quantum physics, laser optics, nonlinear optics, optical fibers, ocean engineering, and electronic engineering. The physical interpretation of the extracted solutions is visualized in two-dimensional, three-dimensional, and contour graphics based on numerical simulation by using the computer software Mathematica. The presented research will be helpful for further investigation of analytical solitary wave and soliton solutions to the complex, higher order nonlinear evolution equations.

This paper aims to explore the nonlinear dynamics of the well-known nonlinear partial differential equations, namely, Estevez–Mansfield–Clarkson (EMC) and Sharma–Taso–Olver (STO) equations. The presented models have useful applications in various fields. The EMC equation clarifies the complex dynamics of waves in shallow water and fluid physics. In nuclear physics, the STO model is pertinent to particle fission and fusion processes. This work offers Riccati sub-equation neural networks to provide exact solutions for space–time partial differential equations. The proposed method incorporates the solutions of the Riccati problem into neural networks. Neural networks are multi-layer computer models with activation functions and weight functions that connect neurons across the input, hidden, and output layers. In this approach, each neuron in the first hidden layer is assigned to the solutions of the Riccati equation. Consequently, the new trial functions are established. The proposed method provides exact solutions to the studied models in the forms of bright, dark, singular, combined, and complex solitons. Moreover, generalized hyperbolic function solutions, trigonometric function solutions, and generalized rational solutions are also recovered. This study introduces innovative solutions as the proposed methodology is used in the neural network model. A variety of graphs have been sketched for the physical behavior of the obtained solutions. By establishing the dependability of the method used, this research’s outcomes could advance our grasp of nonlinear behavior in targeted systems.

In this work, we study the solitary wave profiles of the fractional-Sharma–Tasso–Olver equation, which is applicable to particle fission and fusion mechanisms in nuclear physics. In numerical and analytical theories, exact solitary wave solutions are of the uttermost importance for such equations. Improved analytical methods are essential for a deeper understanding of dynamics, despite their widespread implementation. In this study, we use the advanced analytical techniques known as generalized Arnous method, modified generalized Riccati equation mapping technique, and Riccati extended simple equation approach for securing a variety of solutions. This study marks a significant milestone by applying the prescribed techniques to the proposed equation using truncated M-fractional derivatives and providing a significant contribution to the existing literature. This equation is widely regarded as a model that illustrates the propagation of nonlinear dispersive waves in inhomogeneous media. Using the suitable wave transformation with the fractional-derivative, the governing equation is converted into an ordinary differential equation to get the required solutions. Various types of solutions, such as mixed, dark, singular, bright–dark, bright, complex, and combined solitons, are extracted. Moreover, another important aspect of this study is to discuss the multistability and sensitivity analysis of the studied model by the assistance of the Galilean transformation and perturbation term. The utilized methods have strong computing capacity, which helps them effectively handle the exact solutions with high accuracy in these systems. In addition, we depict 3D and 2D phase portrait graphs with appropriate parameters to illustrate the solution’s behavior.

Abstract This work examined solitary wave solutions to the nonlinear damped Korteweg–de Vries equation by employing the new auxiliary equation approach. The physical structure to the secured solutions visualized in dark solitons, bright solitons, periodic solitons, kink and anti-kink wave solitons, peakon bright and dark solitons, and dispersive solitary waves. The physical interpretation of constructed solutions is visually portrayed using two-dimensional, three-dimensional, and contour plots on the basis of numerical simulation, which help comprehend the physical features of nonlinear behaviour for the solitary waves. The explored solutions will be play important role in Mathematical physics, ion-acoustic waves, dust-acoustic waves, and plasma physics. This study has demonstrated that our suggested method is more beneficial, successful, strong and effective for studying analytically various nonlinear partial differential equations (NLPDEs) that arise in mathematical physics, engineering, plasma physics, and many other scientific fields.