Research on Analytical Solution, Dynamic Analysis and Port Engineering Application of the Simplified Modified Camassa-Holm Equation (Smch) Under Fractional Derivatives

To accurately characterize the nonlinear dynamic behavior of shallow water waves, this study focuses on the Simplified Modified Camassa-Holm (SMCH) equation with Conformable Fractional Derivative (CFD), and applies the extended tanh-function method to the analytical solution of this equation for the first time in our best knowledge. Through traveling wave transformation and homogeneous balance method, various analytical solutions are derived, including single-peak solitons, composite double-peak solitons, single-peak with shoulder solitons, and periodic soliton trains. Using MATLAB, 3D and 2D visualization models are constructed to systematically analyze the regulatory mechanisms of fractional order α and free parameter ω on wave field propagation characteristics, energy distribution, and morphological stability. To verify the effectiveness of the model, this study establishes a comparative system consisting of various fractional derivative forms, which confirms that the CFD has distinct advantages in local detail simulation and computational stability. Combining phase diagrams and Lyapunov exponents, the evolution law of the equation from periodic motion to chaotic state is revealed for the first time. Two numerical methods, fourth-order Runge-Kutta (RK44) and sixth-order accurate Huta86, are used to verify the reliability of the analytical solutions. The maximum absolute error of Huta86 is less than 2×10⁻¹², which is two orders of magnitude more accurate than RK44. The research results improve the solution system of fractional shallow water wave equations and provide accurate theoretical support and parameter optimization basis for wave load prediction and protective structure design in port and coastal engineering.