# Normal form coordinates for the KdV equation having expansions in terms of pseudodifferential operators

## TL;DR

This paper constructs real analytic, canonical coordinates for the KdV equation near finite gap potentials, using pseudodifferential and paradifferential operators to facilitate stability analysis under perturbations.

## Contribution

It introduces a novel coordinate system for KdV involving pseudodifferential operators, enabling normal form transformations and stability studies.

## Key findings

- Coordinates are pseudodifferential operators with Fourier principal part.
- Hamiltonian is in normal form up to order three.
- Hamiltonian vector field admits paradifferential expansion.

## Abstract

Near an arbitrary finite gap potential we construct real analytic, canonical coordinates for the KdV equation on the torus having the following two main properties: (1) up to a remainder term, which is smoothing to any given order, the coordinate transformation is a pseudodifferential operator of order 0 with principal part given by the Fourier transform and (2) the pullback of the KdV Hamiltonian is in normal form up to order three and the corresponding Hamiltonian vector field admits an expansion in terms of a paradifferential operator. Such coordinates are a key ingredient for studying the stability of finite gap solutions of the KdV equation under small, quasi-linear perturbations.