# Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions

## TL;DR

This paper develops a method to transform nonlinear, non-autonomous PDEs into a normal form in infinite-dimensional spaces, enabling clearer analysis of invariant manifolds and approximate solutions.

## Contribution

It introduces a normal form approach for nonlinear, non-autonomous PDEs viewed as ODEs in Fréchet spaces, facilitating the study of invariant and approximate center manifolds.

## Key findings

- Normal form transformation separates center, stable, and unstable coordinates.
- Method applies to a class of nonlinear, non-autonomous PDEs.
- Enables analysis of approximate center manifolds with controlled precision.

## Abstract

We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ODE are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. In particular, our method empowers us to study approximate centre manifolds, given by solutions of ODEs that are central up to a desired, possibly nonzero precision. The main motivation is the case where the Fr\'echet space in question is a suitable function space, and the maps involved in an ODE in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ODE is a finite-dimensional PDE. We show that our methods apply to a relevant class of nonlinear, non-autonomous PDEs in this way.