Derivation of normal forms for dispersive PDEs via arborification

TL;DR

This paper introduces a systematic method for deriving normal forms of dispersive PDEs using decorated trees and Hopf algebra morphisms, connecting algebraic structures with dispersive equation analysis.

Contribution

It presents a novel application of arborification maps from Hopf algebra theory to the derivation of normal forms for dispersive PDEs, bridging algebra and PDE analysis.

Findings

Hopf algebra structures naturally appear in dispersive PDE analysis

Arborification maps facilitate the decomposition of dispersive equations

The approach connects dynamical systems techniques with PDE normal form theory

Abstract

In this work, we propose a systematic derivation of normal forms for dispersive equations using decorated trees introduced in arXiv:2005.01649. The key tool is the arborification map which is a morphism from the Butcher-Connes-Kreimer Hopf algebra to the Shuffle Hopf algebra. It originates from Ecalle's approach to dynamical systems with singularities. This natural map has been used in many applications ranging from algebra, numerical analysis and rough paths. This connection shows that Hopf algebras also appear naturally in the context of dispersive equations and provide insights into some crucial decomposition.