Multi-soliton solutions of the <i>x</i>-dependent variable-coefficient KdV equation via the hirota method

Abstract We first apply the Weiss–Tabor–Carnevale Painlevé analysis to a spatially varying coefficient Korteweg–de Vries (KdV) equation with power-law type coefficients a ( x ) = αx m and b ( x ) = βx 3 m . The principal branch is found to have leading exponent p = − 2, resonance indices r = − 1, 4, 6, and no compatibility obstructions at the positive resonances, indicating that the family passes the Painlevé test and is a strong candidate for analytic integrability. Building on this integrability result, this paper presents a comprehensive study on the exact two- and three-soliton solutions of a generalized KdV equation with time-dependent coefficients using the Hirota bilinear method. The analytical solutions are derived explicitly and validated via symbolic computation in Mathematica . Detailed visualizations are provided to illustrate the propagation and interaction behavior of multi-soliton structures. Furthermore, a physical interpretation of soliton collisions is discussed based on surface plots and animated simulations. These results demonstrate the robustness of the Hirota method in modeling nonlinear wave phenomena in variable coefficient systems.