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Derivation of normal forms for dispersive PDEs via arborification

Yvain Bruned

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.

Derivation of normal forms for dispersive PDEs via arborification

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.
Paper Structure (4 sections, 7 theorems, 97 equations)

This paper contains 4 sections, 7 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Shuffle Hopf algebra
  3. Fourier decorated trees
  4. Normal forms

Key Result

Proposition 2.1

The map $X_{st}$ is a character on the shuffle Hopf algebra $T(A)$.

Theorems & Definitions (9)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 4.1
  • Theorem 4.2