In this study, we investigate the \((2+1)\)-D Jaulent-Miodek (JM) equation, which is significant due to its energy-based Schrödinger potential and applications in fields such as optics, soliton theory, signal processing, geophysics, fluid dynamics, and plasma physics. Given its broad utility, a rigorous mathematical analysis of the JM equation is essential. The primary objective of this work is to derive exact soliton solutions using the Modified Sub-Equation (MSE) and Modified Auxiliary Equation (MAE) techniques. These solutions are computed using Maple 18, and encompass a variety of wave structures, including bright solitons, kink solitons, periodic waves, and singular solitons. The potential applications of these solutions span diverse domains, such as nonlinear dynamics, fiber optics, ocean engineering, software engineering, electrical engineering, and other areas of physical science. Through numerical simulations, we visualize the physical characteristics of the obtained soliton solutions using three distinct graphical formats: 3D surface plots, 2D contour plots, and line plots, based on the selection of specific parameter values. Our results demonstrate that the MSE and MAE techniques are not only efficient but also straightforward in extracting soliton solutions for the JM equation, outperforming other existing methods. Furthermore, the solutions presented in this study are novel, representing contributions that have not been previously reported in the literature.

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.