Analysis of shallow water waves using a nonlinear conformable M-fractional Boussinesq–Kaup system

In this paper, we present and explore a nonlinear conformable M-fractional extension of the Boussinesq-Kaup system to examine the dynamics of shallow water waves. The proposed nonlinear conformable M-fractional system is considered an extension of the ferromagnetic spin chain equation, and it is able to describe the complicated dynamics of wave propagation in the presence of nonlinearity and memory effects. The suggested model is developed as a system of nonlinear partial differential equations with coupled conformable M-fractional derivatives providing an efficient way to consider nonlinearity, memory effects, and dispersal of waves. The direct algebraic method is an efficient but a simple method to deal with such complex nonlinear systems providing the exact traveling wave solutions. The inferred solutions of the analysis are then discussed by way of graphical representations with a view of explaining their physical meaning. In order to find out qualitative dynamics of the system, the analysis of phase portraits is conducted, which gives the opportunity to characterize equilibrium points in detail. The paper analyzes kink soliton dynamics in the conformable KPP equation, highlighting the impact of fractional derivatives on wave evolution. By employing bifurcation analysis, the research illustrates how sensitive wave patterns are to subtle parameter variations and initial conditions. These results mirror the complexity of actual shallow water waves. The integrated approach-combining fractional calculus with nonlinear and graphical analysis-provides a powerful tool for interpreting wave phenomena, confirming the methodology's utility in exploring diverse nonlinear dynamics.