Slowly varying solitary waves in the extended shallow water model with a perturbation term

Abstract The extended Korteweg–de Vries (KdV) equation includes higher-order dispersive and nonlinear terms compared to the classical KdV equation, making it more aligned with real-world shallow water models. Mathematically, the expanded KdV equation in this paper refers specifically to the third equation of the KdV hierarchy. Taking into account the dissipative effects present in actual physical processes, this study investigates the slowly varying solitary waves of the perturbed extended KdV equation. By employing the multiple-scale method, we derived an approximate analytical solution for this perturbed system. Subsequently, we obtained initial conditions with physical significance and used the Fourier spectral method to acquire numerical solutions for the slowly varying solitary waves, which provides a mathematical basis for subsequent fluid dynamics research.