Qualitative dynamics and solitary wave phenomena in a new extended Boussinesq-Type KdV equation

• A novel Boussinesq-type Korteweg-de Vries (KdV) equation is investigated. • The Boussinesq-type KdV equation, which generalizes the classical KdV equation, is frequently employed in plasma physics to model wave phenomena in elastic media, shallow water waves, and other systems exhibiting both propagation and dispersion characteristics. • Deriving analytical solutions of the suggested equation using the Kumar Malik approach equation. • All analytical results are verified using the Mathematica software package. In the present study, the extended Boussinesq-type Korteweg-de Vries (KdV) equation is solved by the Kumar-Malik (KM) method to yield new families of analytical traveling-wave solutions. The model has been transformed into ordinary differential equations through suitable traveling-wave transformations, and the ODE is solved to obtain explicit solutions as Jacobi elliptic, hyperbolic, trigonometric, and exponential functions. The dynamical characteristics of these solutions are further examined via bifurcation analysis, sensitivity to initial conditions, and phase-portraits. In addition, the chaotic dynamics of the system are explored using bifurcation diagrams, Poincaré maps, and time series representations. The results demonstrate multifaceted dynamical effects, including periodic, quasi-periodic, and chaotic transitions. The study provides valuable information on the nonlinear complex processes of extended Boussinesq-type KdV models and may serve as a basis for additional theoretical and applied research.