Birkhoff normal form for some nonlinear PDEs
type: Item
instance of: scholarly article
title: Birkhoff normal form for some nonlinear PDEs
zbMATH Open document ID: 1032.37051
author: Dario Bambusi
published in: Communications in Mathematical Physics
publication date: 2003-08-17T00:00:00Z
review text: Infinite-dimensional Hamiltonian systems are considered, with Hamiltonian given by \(H=h_0+f\). Here \(h_0(p, q)=\sum_j\omega_j (p_j^2+q_j^2)/2\), and \(f\) is of order at least three in \((p, q)\). Under certain assumptions on the frequencies (described in more detail below) it is shown that there is a canonical transformation \({\mathcal T}\) defined in a neighborhood of the origin \((p, q)=(0, 0)\) such that \(H\circ {\mathcal T}=h_0+Z+R\), where \(Z\) is in Birkhoff normal form and \(R\) is a remainder term of an arbitrary high order that can be prescribed. The proof uses an inductive argument which goes roughly as follows: If a transformation \({\mathcal T}^{(k)}\) is already known that puts \(H\) into the form \(H\circ {\mathcal T}^{(k)}=h_0+Z^{(k)}+R^{(k)}\), then basically speaking the remainder term \(R^{(k)}\) is split as \(R^{(k)}=R^{(k)}_{\text{low}, \leq 2}+R^{(k)}_{\text{low}, \geq 3} +R^{(k)}_{\text{high}}\). Here \(R^{(k)}_{\text{high}}\) is of sufficiently high order in \((p, q)\), \(R^{(k)}_{\text{low}, \leq 2}\) contains the lower order terms that are of order \(\leq 2\) in \((p_j, q_j)\) with \(|j|\geq N+1\) (\(N\) fixed), whereas \(R^{(k)}_{\text{low}, \geq 3}\) denotes the lower order terms that are of order \(\geq 3\) in \((p_j, q_j)\) with \(|j|\geq N+1\). This spatial and frequency cutoff at order \(N\) allows to prove that the contribution of \(R^{(k)}_{\text{low}, \geq 3}\) to the flow is `negligible' if all parameters (including \(N\)) are combined in an appropriate way. The term \(R^{(k)}_{\text{low}, \leq 2}\) can be put into normal form by introducing a suitable transformation \({\mathcal T}_k\) via a vector field \(\chi_k\) for whose definition small denominators occur in a natural way. It turns out that then, defining \({\mathcal T}^{(k+1)} ={\mathcal T}^{(k)}\circ {\mathcal T}_k\), the Hamiltonian can be transformed as \(H\circ {\mathcal T}^{(k+1)}=h_0+Z^{(k+1)}+R^{(k+1)}\) which is at least one order better than \(H\circ {\mathcal …
Mathematics Subject Classification ID: 37K55
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 70H15
zbMATH DE Number: 1964918
zbMATH Keywords: Birkhoff normal form
zbMATH Keywords: small denominators
zbMATH Keywords: nonlinear wave equation
reviewed by: Markus Kunze
MaRDI profile type: Publication
full work available at URL: s00220-002-0774-4
OpenAlex ID: W1977120242
DOI: 10.1007/S00220-002-0774-4
description: scientific article; zbMATH DE number 1964918