The transformation of water waves in shallow regions is a complex phenomenon. As waves reach the beach, they undergo many processes, including shoaling, refraction, diffraction, and breaking. The importance of precisely forecasting wave patterns in shallow water areas is underscored by their influence on sediment transport, circulation, and other nearshore processes. This work looks at the one-dimensional modified spectral Korteweg–de Vries (mSKdV) equation as a mathematical framework to improve the comprehension of nonlinear wave transformation in shallow water. The research provides insights into the influence of changing bathymetry and nonlinearity on wave evolution along the beach. The unified method and the \(\left(\frac{{G}^{\prime}}{G}, \frac{1}{G}\right)\) method are two effective techniques that are utilized in order to derive soliton solutions for this equation. A bifurcation analysis is conducted to ascertain the phase portrait of this equation. Beyond the direct observations, a linear stability analysis was also performed. This crucial step further confirms and quantifies how the stability of our solutions changes across various bathymetries. This analysis provides a rigorous mathematical framework, reflecting the solutions' resilience and behavior under different underwater topographies. A synthesis of three-dimensional plots, modified two-dimensional plots concerning the time variable t, and density plots is employed to illustrate the precise solutions obtained. These diagrams illustrate the transition from the shallow water zone to the shoreline. Various soliton phenomena have been observed, including periodic solitons, bright and dark solitons, periodic breathers, quasi-breathers, breathers, and kinks. This discovery significantly advances the development of soliton solutions for the mSKdV equation. It also offers significant insights on the spectrum wave height and setup that unfold in coastal engineering, especially in regions with shallow seas.
