Dynamics study of stability analysis, sensitivity insights and precise soliton solutions of the nonlinear (STO)-Burger equation

The purpose of this manuscript is to investigate the precise soliton solution of the \((1 + 1)\)-dimensional Sharma–Tasso–Olver–Burgers equation. This study thoroughly explores the characteristics and solutions of this nonlinear equation. To pinpoint the exact soliton solution, we employ analytical techniques, specifically the \(\exp (-\phi (\xi ))\)-expansion method, as well as stability and sensitivity analysis. We begin by introducing the equation and discussing its role in mathematical modeling. It’s worth noting that the equation’s inherent nonlinearity adds complexity and poses challenges for analysis and solution. Our research objective is to identify solutions through graphical interpretation, allowing us to gain insights into their behavior. To achieve this, we utilize relevant parameter values to emphasize the physical properties of the provided data. We use the Mathematica and Maple software to demonstrate the solutions that have been discovered in 3D, 2D, and contour plots for the purpose of physical expression and graphical representation. The investigated system is shown to be stable since a little change in the initial conditions does not result in an immense shift in solutions. The sensitivity and stability analyses are also provided at various initial conditions. Importantly, these methods have broader applications and are valuable for conveying nonlinear physical models in the field of nonlinear sciences, not limited to the Sharma–Tasso–Olver–Burgers equation alone.