Perturbation of the nonlinear Schrödinger equation by a localized nonlinearity

We revisit the perturbative theory of infinite dimensional integrable systems developed by P. Deift and X. Zhou \cite{DZ-2}, aiming to provide new and simpler proofs of some key $L^{\mathrm{\infty }}$ bounds and $L^p$ \emph{$\mathit{\text{a}}\mathit{\text{priori}}$} 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.