The problem of integrable discretization: Hamiltonian approach
type: Item
instance of: scholarly article
title: The problem of integrable discretization: Hamiltonian approach
zbMATH Open document ID: 1033.37030
published in: Progress in Mathematics
publication date: 2003-07-27T00:00:00Z
review text: This book is devoted to the problem of integrable discretization in the field of integrable systems. More concretely, it deals with the problem: how to discretize a given finite-dimensional integrable system, keeping its integrability property as in the theories of symplectic numerical algorithms and integrable couplings. The problem can be precisely described as follows: Given an integrable Hamiltonian system \(\dot x =f(x)=\{H,x\}\) on an \(N\)-dimensional Poisson manifold \(\mathcal P\) with a Poisson bracket \(\{\cdot,\cdot\}\), which possesses \(N\) functionally independent integrals of motion \(I_k(x)\), \(1\leq k\leq N,\) in involution under the Poisson bracket \(\{\cdot,\cdot\}\), find a one-parameter family of diffeomorphisms from \(\mathcal P\) to \( \mathcal P\): \[ \tilde x =\Phi(x;h),\quad h \text{ being a small positive parameter}, \] which satisfies \(\Phi(x,h)=x+hf(x)+\text{O}(h^2)\) and \(\{\tilde x,\tilde y\}_h= \{x,y\}_h\) under some Poisson bracket \(\{\cdot,\cdot\}_h\) with \(\{\cdot,\cdot\}_h=\{\cdot,\cdot\}+\text{O}(h)\), and possesses \(N\) functionally independent invariant functions \(I_k(x,h)\), \(1\leq k\leq N,\) in involution under the Poisson bracket \(\{\cdot,\cdot\}_h\) and with \(I_k(x,h)=I_k(x)+\text{O}(h)\), \(1\leq k\leq N\). The book consists of three parts. Part I ``General Theory'' has two chapters. The first chapter reviews the fundamental concepts and techniques in the Hamiltonian theory on Poisson manifolds, which include Poisson bracket, Hamiltonian flow, bi-Hamiltonian system, Euler-Lagrangian equations on Lie groups, and Lagrangian reductions and Euler-Poincaré equations. The second one provides a compact exposition of the \(r\)-matrix theory containing linear and quadratic \(r\)-matrix structures and related \(R\)-operators. It then presents a general recipe for constructing an integrable discretization, through making Bäcklund transformations close to the identity transformation within the \(r\)-matrix theory. The rest …
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37M15
Mathematics Subject Classification ID: 70H06
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 1953454
zbMATH Keywords: Integrable lattice
zbMATH Keywords: \(r\)-matrix hierarchy
zbMATH Keywords: Lax representation
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: Integrable discretization
author: Yuri B. Suris
reviewed by: Wen-Xiu Ma
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1953454