We revisit the perturbative theory of infinite-dimensional integrable systems developed by P. Deift and X. Zhou, aiming to provide new and simpler proofs of some key L∞ bounds and Lᵖ a priori estimates. Our proofs emphasize a further step towards understanding focusing problems and extend the applicability to other integrable models. As a concrete application, we examine the perturbation of the one-dimensional defocusing cubic nonlinear Schrödinger equation and modified KdV equations.