Birkhoff normal form via decorated trees

2025/05/07 by Jacob Armstrong-Goodall, Armstrong-Goodall, Jacob, Yvain Bruned +1
Computer Science · Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations #Polynomial and algebraic computation #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2505.04547

openalex publication_date 2025/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive an explicit tree based ansatz for the Birkhoff normal form up to any order in the context of Hamiltonian PDEs. To do so we make use of a tree based representation of iterated Poisson brackets to encode the nested Taylor expansions along flows of a sequence of symplectic transformations. As an example we consider the cubic Schrödinger equation.

Citations

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Related

• Birkhoff Normal Form and Hamiltonian PDEs
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• Analytic Extension of the Birkhoff Normal Forms for the Free Rigid Body Dynamics on SO(3)