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Birkhoff normal form via decorated trees

Jacob Armstrong-Goodall, Yvain Bruned

TL;DR

Addresses the problem of obtaining an explicit Birkhoff normal form for Hamiltonian PDEs. Introduces a decorated-tree formalism to encode the nested iterated Poisson brackets generated by a sequence of symplectic transformations used to remove non-resonant interactions. The main contribution is a closed-form, order-by-order decomposition of the reduced Hamiltonian into sums over decorated trees, plus a recursive description of the generating functions that drive the conjugations. The results illuminate the combinatorial structure of resonances in Hamiltonian PDEs and provide a constructive framework that could enhance long-time dynamics analysis and numerical methods, with the cubic Schrödinger equation as a guiding example.

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.

Birkhoff normal form via decorated trees

TL;DR

Addresses the problem of obtaining an explicit Birkhoff normal form for Hamiltonian PDEs. Introduces a decorated-tree formalism to encode the nested iterated Poisson brackets generated by a sequence of symplectic transformations used to remove non-resonant interactions. The main contribution is a closed-form, order-by-order decomposition of the reduced Hamiltonian into sums over decorated trees, plus a recursive description of the generating functions that drive the conjugations. The results illuminate the combinatorial structure of resonances in Hamiltonian PDEs and provide a constructive framework that could enhance long-time dynamics analysis and numerical methods, with the cubic Schrödinger equation as a guiding example.

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.
Paper Structure (5 sections, 3 theorems, 77 equations)

This paper contains 5 sections, 3 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Hamiltonian Equations
  3. The Cubic Schrödinger Equation
  4. The Birkhoff Normal Form Reduction
  5. Decorated trees formulation

Key Result

Lemma 3.2

For $a\geq 0$ and $p\geq\frac{1}{2}$, the space $\ell^{a,p}_b$ is a Hilbert algebra with respect to the convolution of sequences, and where $c$ depends on $p$ only.

Theorems & Definitions (13)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3: Hamiltonain Taylor Expansion
  • Definition 2.5
  • Definition 3.1: Sobolev Norm on Sequences
  • Lemma 3.2
  • Definition 3.3: Truncation Term
  • Remark 3.4
  • example 1
  • Definition 3.5: Truncated Normal Form
  • ...and 3 more