Birkhoff normal form via decorated trees

TL;DR

This paper introduces a tree-based method to explicitly compute the Birkhoff normal form for Hamiltonian PDEs, exemplified by the cubic Schrödinger equation, using iterated Poisson brackets and symplectic transformations.

Contribution

It presents a novel tree-based ansatz for deriving Birkhoff normal forms at any order in Hamiltonian PDEs, enhancing computational clarity and explicitness.

Findings

Explicit tree-based representation for Birkhoff normal form

Application to cubic Schrödinger equation

Framework for nested Poisson brackets

Abstract

We derive an explicit tree based ansatz for the Birkhoff normal form up to any order in the context of Hamiltonian PDEs. To do so we make use of a tree based representation of iterated Poisson brackets to encode the nested Taylor expansions along flows of a sequence of symplectic transformations. As an example we consider the cubic Schr\"odinger equation.