# Rational normal forms and stability of small solutions to nonlinear Schr\"odinger equations

## TL;DR

This paper introduces rational normal forms for nonlinear Schrödinger equations on the circle, enabling long-time control of solutions' Sobolev norms and demonstrating local stability for small initial data.

## Contribution

It develops a new rational normal form technique for NLS equations, allowing for extended stability and integrability results without external parameters.

## Key findings

- Flow conjugated to an integrable system up to small remainder
- Control of Sobolev norms over long timescales
- Local stability of solutions under small perturbations

## Abstract

We consider general classes of nonlinear Schr\"odinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\varepsilon$, for generic initial data of size $\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\varepsilon^{M+1}$. This implies that for such initial data $u(0)$ we control the Sobolev norm of the solution $u(t)$ for time of order $\varepsilon^{-M}$. Furthermore this property is locally stable: if $v(0)$ is sufficiently close to $u(0)$ (of order $\varepsilon^{3/2}$) then the solution $v(t)$ is also controled for time of order $\varepsilon^{-M}$.