Perturbation of the nonlinear Schr\"odinger equation by a localized nonlinearity

TL;DR

This paper improves the theoretical understanding of how localized nonlinear perturbations affect the long-term behavior of the defocusing cubic nonlinear Schrödinger equation, extending integrable systems analysis with simpler proofs and broader applicability.

Contribution

It provides new, simplified proofs of key bounds in perturbative theory and extends the analysis to localized nonlinear perturbations of integrable models.

Findings

Perturbed NLS exhibits the same long-time behavior as the unperturbed equation.

New estimates control the influence of localized higher-order nonlinear terms.

The approach extends to other integrable models.

Abstract

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 $L^\infty$ bounds and $L^p$ \emph{\textit{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 Schr\"odinger 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 Schr\"odinger equation.