Investigating fluctuation varieties in the propagation of the perturbed KdV equation with time-dependent perturbation coefficient

This study investigates the perturbed Korteweg–de Vries equation modified by incorporating time-dependent perturbation coefficient to model random fluctuations within the wave dynamics. This enhanced equation captures the probabilistic aspects of wave behavior in uncertain environments, accounting for the effects of inherent noise. The Hirota bilinear method, tanh-expansion approach, and the sine(cosine)-function method are employed to derive perturbed soliton solutions. By assigning various functional forms such as periodic, polynomial, and decaying exponential, to the proposed time-dependent coefficient, novel solitary wave patterns of types like-breather, regular(singular)-bell shaped, and periodic solutions are emerged with fluctuations. These findings are relevant for systems where environmental variability or intrinsic noise significantly affects dynamics, such as diffusion processes in physics and uncertainty behavior of water waves.