The soliton correlation matrix and the reduction problem for integrable systems
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 \(\psi(\lambda_ i)\) and \(\psi(\mu_ i)\). In the subsequent section the authors consider the reduced systems obtained from (A) by the application of certain additional constraints corresponding to invariance under finite automorphism groups as done by Mikhajlov [Physica 3D, 73 (1981)]. The authors show that the soliton correlation matrix can be used to obtain the solutions of the reduced systems. As examples they give the multisoliton solutions for the SL(n,R)/SO(n) sigma model and the two- dimensional non-abelian Toda lattice. This is an interesting paper. The notion ''soliton correlation matrix'' introduced in this paper should be a valuable contribution to the dressing method.
scientific article; zbMATH DE number 3876262