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.

翻译：本文提出了一种系统推导色散方程正规形的方法，该方法利用了arXiv:2005.01649中引入的装饰树。核心工具是树状化映射，该映射是从Butcher-Connes-Kreimer Hopf代数到Shuffle Hopf代数的态射。它起源于Ecalle处理具有奇点的动力系统的方法。这一自然映射已被广泛应用于代数、数值分析和粗糙路径等多个领域。该关联表明，Hopf代数也自然地出现在色散方程的语境中，并为理解一些关键分解提供了洞见。