The soliton correlation matrix and the reduction problem for integrable systems
type: Item
instance of: scholarly article
title: The soliton correlation matrix and the reduction problem for integrable systems
zbMATH Open document ID: 0549.58025
DOI: 10.1007/BF01218638
author: J. Harnad
author: Yvan Saint-Aubin
author: Steven Shnider
published in: Communications in Mathematical Physics
publication date: 1984-01-01T00:00:00Z
review text: Let (A) \(\psi_{\xi}=U\psi\), \(\psi_{\eta}=V\psi\) be a pair of linear differential equations, where \(U(\lambda,\xi,\eta)\), \(V(\lambda,\psi,\eta)\) and \(\psi(\lambda,\xi,\eta)\) are \(n\times n\) matrices depending on a complex parameter \(\lambda\). Assume that U and V are meromorphic in \(\lambda\). Many nonlinear differential equations, including the celebrated KdV, sine-Gordon equations and various 2- dimensional field equations, are known to be the integrability condition of the above pair of equations, i.e., \(U_{\eta}-V_{\xi}+[U,V]=0\). A matrix \(\chi(\lambda,\xi,\eta)\), called the dressing matrix, is defined by the requirement that the new matrix \({\hat\psi}=\chi\xi\) satisfies the pair of equations \({\hat\psi}_{\xi}=\hat U{\hat\psi },{\hat\psi}_{\eta}=\hat V{\hat\psi }\), where \(\hat U\) and \(\hat V\) possess the same analytic structures as U and V, that is, the same poles and multiplicities. It is assumed that \(\chi\) and \(\chi^{-1}\) are meromorphic with K simple poles in \(\lambda\). The authors introduce an n\(K\times nK\) matrix M, called the soliton correlation matrix, whose ij- th \(n\times n\) block \(M_{ij}\) is defined by \[ M_{ij}=(1/(2\pi i)^ 2)\oint_{\mu_ i}d\mu\oint_{\lambda_ j}d\chi(\chi^{- 1}(\mu)\chi(\lambda))/(\mu -\lambda),\quad i,j=1,...,K, \] where the integrations are on small contours containing only the poles indicated. The authors show that the M-matrix is subjected to the constraints \(\mu_ iM_{ij}-\lambda_ jM_{ij}=- \sum^{K}_{km}M_{ik}M_{mj}\) and it satisfies the following Riccati systems \[ (B)\quad M_{\xi}=p^+M-Ms^+-Mr^+M,\quad M_{\eta}=p^-M-Ms^--Mr^-M, \] where \(p^{\pm},s^{\pm}\) and \(r^{\pm}\) are defined explicitly by matrices U and V. The geometrical interpretation of this matrix Riccati system, formulated previously by the authors [Commun. Math. Phys. 92, 329-367 (1984; Zbl 0532.58036)], is briefly mentioned. Furthermore the authors give the general solution of the system (B) explicitly in terms of \(\psi(\l…
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 22E70
zbMATH DE Number: 3876262
zbMATH Keywords: soliton
zbMATH Keywords: integrable system
zbMATH Keywords: multi-Bäcklund transformation
zbMATH Keywords: sine-Gordon equations
zbMATH Keywords: 2-dimensional field equations
zbMATH Keywords: dressing matrix
zbMATH Keywords: soliton correlation matrix
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01218638
OpenAlex ID: W2027775091
description: scientific article; zbMATH DE number 3876262