Analytical and Numerical Soliton Solutions of the Shynaray II‐A Equation Using the G′G,1G$$ \left(\frac{G^{\prime }}{G},\frac{1}{G}\right) $$‐Expansion Method and Regularization‐Based Neural Networks

ABSTRACT Nonlinear differential equations play a fundamental role in modeling complex physical phenomena across solid‐state physics, hydrodynamics, plasma physics, nonlinear optics, and biological systems. This study focuses on the Shynaray II‐A equation, a relatively less‐explored parametric nonlinear partial differential equation that describes nonlinear wave propagation and soliton dynamics in diverse physical systems. To address the gap in analytical and numerical treatments of this equation, we derive new exact soliton solutions using the ‐expansion method, yielding exponential, trigonometric, and rational solution forms. Moreover, we introduce a novel hybrid framework by applying regularization‐based backpropagation neural networks to approximate these solutions numerically. This dual approach, combining symbolic analysis with machine learning, offers a robust means to validate and extend analytical findings. The originality of this work lies in presenting, for the first time, a unified analytical‐neural network framework for solving the Shynaray II‐A equation, which has not been systematically studied in prior literature. Comparative error metrics and visualization confirm the accuracy and efficiency of the proposed method, providing new insights into the behavior of nonlinear wave systems and demonstrating the potential of hybrid soliton modeling strategies.