Dynamics of waveforms in the potential KdV equation incorporating time-dependent perturbation coefficient

In this study, we extend the potential Korteweg–de Vries equation by incorporating a time-dependent coefficient within its perturbation term. The refined model captures how solitary waves, such as water waves, optical signals, and photons, evolve under different perturbation coefficients. To investigate its dynamics, two main directions are pursued. First, several analytical techniques, including the Hirota bilinear method, the rational sine–cosine function method, and the rational sinh–cosh function method, are applied to derive possible soliton solutions. Second, a complementary graphical analysis is conducted to examine the bifurcation behavior of these solutions. This exploration employs diverse functional assignments as test cases, highlighting scenarios of periodic, growing, and decaying perturbations. To the best of available literature, the qualitative classification of propagation behavior under different perturbation profiles is explored here for the first time.The findings of the current work are significant for systems where environmental variability or intrinsic factors play a critical role, such as signal processing, fluid dynamics, optical wave transmission, and the unpredictable dynamics of water waves. The capacity to control or reduce fluctuations by employing specific forms of the perturbation coefficient offers crucial insights for developing systems and technologies that demand consistent and reliable wave propagation, even under complex environmental conditions.