On the higher Poisson structures of the Camassa-Holm hierarchy
type: Item
instance of: scholarly article
title: On the higher Poisson structures of the Camassa-Holm hierarchy
zbMATH Open document ID: 1165.37030
author: G. Ortenzi
author: Marco Pedroni
author: Volodya Rubtsov
published in: Acta Applicandae Mathematicae
publication date: 2008-09-02T00:00:00Z
review text: The article under review is to generate higher-order Poisson structures related to the Camassa-Holm equation. The key point is to adopt the squared solutions to a linear second-order differential equation associated with a linear combination of the Hamiltonian pair. Then the authors represent the generating series of higher-order Poisson structures as a compact formula using the squared solutions and the inverse operator \(\partial ^{-1}\). The article is clearly written and only the domain and range space for the inverse operator \(\partial ^{-1}\) is not sufficiently treated. In particular, the condition \(\lim _{x\to \infty}\xi (x) =-\int _{x\to -\infty} \xi (x) \in \mathbb{R} \) for functions in the space requires \(\int_{-\infty}^\infty \xi (y) \, dy =0\) to guaranntee the convergence of the two integrals \(\int_{-\infty}^x \xi (y) \, dy \) and \(\int_{-\infty}^x \xi (y) \, dy \). This makes that the inverse operator \(\partial ^{-1}\) goes out into another space.
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K05
zbMATH DE Number: 5319014
zbMATH Keywords: Camassa-Holm equation
zbMATH Keywords: Hamiltonian structures
zbMATH Keywords: recursion hereditary operators
zbMATH Keywords: symmetries
zbMATH Keywords: conservation laws
zbMATH Keywords: bi-Hamiltonian structures
reviewed by: Wen-Xiu Ma
MaRDI profile type: Publication
full work available at URL: s10440-008-9188-4
OpenAlex ID: W1998081330
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DOI: 10.1007/S10440-008-9188-4
description: scientific article; zbMATH DE number 5319014