Ming Li

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.