Derivation of normal forms for dispersive PDEs via arborification

Yvain Bruned
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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.
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通过树枝化推导分散 PDE 的正常形式
在这项工作中,我们提出了一种利用 arXiv:2005.01649 中引入的装饰树系统推导分散方程正常形式的方法。关键工具是树化映射,它是从布彻-康涅斯-克里默霍普夫代数到舒弗-霍普夫代数的变形。它源于埃卡勒研究具有奇点的动力系统的方法。这一自然映射被广泛应用于代数、数值分析和粗糙路径等领域。这种联系表明,霍普夫代数也自然地出现在分散方程中,并为一些关键的分解提供了见解。
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