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arXiv · 2508.11463

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

Gong Chen·Jiaqi Liu·Yuanhong Tian
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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.

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Gong Chen, Jiaqi Liu, Yuanhong Tian. 2025-08-15. Perturbation of the nonlinear Schr\"odinger equation by a localized nonlinearity. https://arxiv.org/abs/2508.11463

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