In this article, we investigate the STF modified Benjamin-Bona-Mahony (STF-mBBM) equation, which is important in understanding wave phenomena across various technical scenarios such as ocean waves, acoustic gravity waves and cold plasma physics. We describe the fundamental properties of fractional calculus and its application to the STF-mBBM equation. Utilizing beta derivatives, we enhance our understanding of the intricate wave dynamics involved. Through the modified \(\left( \frac{G'}{G^{2}}\right)\) -expansion method (M \(\left( \frac{G'}{G^{2}}\right)\) -EM), we derive periodic, and kink singular soliton solutions and represent them graphically. We present the influence of the fractional parameter on traveling wave with 2D, 3D, surface and contour plots, providing a thorough understanding of the physical phenomena associated with the fractional model. In addition, we utilize the Hamiltonian property to analyze the chaotic dynamics of the solutions we’ve acquired. We perform two types of analysis using the Galilean transformation: a local sensitivity examination is conducted to see how the model responds to changes in individual input factors, and a global sensitivity examination is conducted to comprehend the correlation between the variability in the results and the variability in each input variable throughout its whole range of significance. This comprehensive approach allows us to determine traveling wave solutions effectively, offering new insights into the non-linear dynamical behavior of the system. The findings from this study are unique and significant for further exploration of the equation, offering valuable insights for future researchers.
