Action-angle maps and scattering theory for some finite-dimensional integrable systems. II: Solitons, antisolitons, and their bound states
type: Item
instance of: scholarly article
title: Action-angle maps and scattering theory for some finite-dimensional integrable systems. II: Solitons, antisolitons, and their bound states
zbMATH Open document ID: 0842.58049
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1996-08-04T00:00:00Z
review text: It is the second part of the author's serial work which present an explicit construction of an action-angle map for the non-relativistic Calogero-Moser systems with \(1/sh^2\) and \(-1/ch^2\) pair potentials, and for relativistic generalizations thereof. There are seven chapters and an appendix in the paper. Chapter 2 is concerned with the determination of the range of variation of the spectrum of the Lax matrix \(L(p)\) as \(p\) varies over the subset \(\Omega^\#_s\) of \(\Omega_s\) where \(L\) has \(\ell\) complex-conjugate pairs of eigenvalues (see paper I). Chapter 3 is concerned with showing that the date set can be promoted to a symplectic manifold. Here is a crucial step towards solving this problem: it enables to reduce the problem to the special (pure soliton) case \(N_+ N_- = 0\) already handled in paper I. In Chapter 4 the algebraic and analytic information assembled in chapters 2 and 3 is combined with geometric information to solve the above problem. The special case \(N_+ N_- =1\) is given in considerable detail at the beginning of chapter 5. Here it also shown that the Lax matrix is not diagonalizable on the exceptional set (separatrix) \(\Omega_e \equiv \Omega/\Omega_s\). In chapter 6 the extension information is used to study the character and temporal asymptotics of a large class of mutually commuting Hamiltonian flows, containing in particular the flows generated by the Hamiltonian (1.1) and (1.2) (see this paper) in the nonrelativistic and relativistic case, resp. Besides the general case, this chapter also deals with a special situation of considerable physical interest. [For Part I see the author, Commun. Math. Phys. 115, No. 1, 127-165 (1988; Zbl 0667.58016), for Part III, see the review below].
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35Q51
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 777765
zbMATH Keywords: symplectic geometry
zbMATH Keywords: action-angle maps
zbMATH Keywords: Calogero-Moser systems
zbMATH Keywords: Lax matrix
author: Simon Ruijsenaars
reviewed by: Li-Ren Sheng
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195164945
OpenAlex ID: W2004223874
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DOI: 10.2977/PRIMS/1195164945
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description: scientific article; zbMATH DE number 777765