# Convergence to normal forms of integrable PDEs

## TL;DR

This paper establishes a method for transforming integrable PDEs like KdV, NLS, and Toda lattice into their normal forms using Poincaré normal form, enabling the construction of Birkhoff coordinates near the origin.

## Contribution

It provides a novel iterative approach to explicitly construct local analytic transformations to normal forms for integrable PDEs in infinite-dimensional spaces.

## Key findings

- Constructed local analytic coordinate transformations for integrable PDEs.
- Applied the method to KdV, NLS, and Toda lattice equations.
- Proved existence of Birkhoff coordinates near the origin.

## Abstract

In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincar\'e normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighbourhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincar\'e normal form and of the transformation to it, through a rapid convergence algorithm.