The study of coherent structures of combined KdV-mKdV equation through integration schemes and stability analysis

The combined Korteweg-de Vries (KdV)-modified KdV (mKdV) equation widely appears as an integrable model in a wide range of interacting physical phenomena to explore their nonlinear dynamical features with geometrical structures. In this work, we present a comprehensive physical analysis of a bunch of solitary waves associated with a combined KdV-mKdV equation, aiming to elucidate the properties and behavior of these waves in the context of propagation in the background of interactions. The study is specifically concerned with analytically examining of combined KdV-mKdV equation, with a focus on algebraic and geometrical properties of solitary wave solutions with their stability analysis. The nature and stability of solitary waves are studied using mathematical methods such as the Kudrayashov methodology and the \(\exp (-\Phi (\zeta ))\) $exp(-\mathrm{\Phi }(\zeta ))$ -expansion method. This investigation tries to shed light on the behavior of solitary waves in various beginning situations and parameter regimes. Plotting 3D surface graphs, contour plots, and line graphs for different parameter values enables the observation of the graphical properties of the developed solutions. The obtained solutions exhibit kink, singular periodic, and periodic behaviors. The answers generated using these two methodologies also show the graphical representations of the soliton propagation.