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：本文提出了一种利用arXiv:2005.01649中引入的装饰树系统推导色散方程正规形的方法。核心工具是树形化映射——一个从Butcher-Connes-Kreimer Hopf代数到Shuffle Hopf代数的态射，该方法源于Ecalle处理具奇点动力系统的理论。这一自然映射已广泛应用于代数、数值分析和粗糙路径等领域。该关联表明Hopf代数在色散方程研究中自然出现，并为若干关键分解提供了理论洞见。