We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou [8], aiming to provide new and simpler proofs of some key \(L^\infty \) bounds and \(L^p\) a priori estimates. Our proofs emphasizes a further step towards understanding focussing problems and extends the applicability to other integrable models. As a concrete application, we examine the perturbation of the one-dimensional defocussing cubic nonlinear Schrödinger equation by a localized higher-order term. We introduce improved estimates to control the power of the perturbative term and demonstrate that the perturbed equation exhibits the same long-time behavior as the completely integrable nonlinear Schrödinger equation.
