We study the dynamics of solitons as solutions to the perturbed KdV (pKdV) equation ∂tu = −∂x(∂ xu + 3u − bu), where b(x, t) = b0(hx, ht), h 1 is a slowly varying, but not small, potential. We obtain an explicit description of the trajectory of the soliton parameters of scale and position on the dynamically relevant time scale δh−1 log h−1, together with an estimate on the error of size h. In a...