Diversity of Solitary Structures by the Application of Symbolic Neural Network-Based Approach: Exploring the Strain Wave Equation

A novel modified generalized Riccati equation mapping neural network-based approach is the basic theme of this study by exploring the nonlinear dynamical characteristics of the the strain wave model’s soliton solutions, which govern wave propagation in micro structured solids. Strain waves are particularly intriguing, since they preserve their form and speed throughout transmission. The nonlinear dynamical behaviors of strain waves may be modeled by partial differential equations in micro structured materials. In the realm of micro structured solids, there exists a class of phenomena that are referred to as micro strain waves. These waves arise in solids possessing intricate internal architectures, including periodic lattices, precisely engineered metamaterials Understanding these waves is key to designing more complex materials and new acoustic technologies. The activation function and the weight function of the neural network are assigned to each input layer, hidden layer and output layer and the neural network itself is a multi-layer computational network. Using the structure of the neural network, every neuron in the first hidden layer is given solutions to the Riccati equation, and the new highly expressive trial functions are generated in a systematic way. In this way, a large variety of exact soliton solutions are obtained, such as bright, dark, kink, and combined solitons as well as periodic and hyperbolic wave profiles. The influence of the essential physical and mathematical parameters is explored systematically using three-dimensional, two-dimensional and contour visualizations, which illustrate how parameter variations lead to changes in the amplitude, shape and stability of the wave structures. The solutions presented reveal the dynamic properties of micro strain solitons which leads to new avenues of investigation in the study of related nonlinear phenomena in micro structured solids. In a broader context, our results highlight the great potential of analytical techniques using neural networks as a powerful and versatile toolset to study complex nonlinear wave models within the applied sciences from acoustics to photonics to smart materials engineering.