Toda lattice hierarchy. II
The \(\ell\)-reduced family, \((TL)_{\ell}\), of the Toda lattice (TL) hierarchy is given. The N-soliton solutions are also constructed by using the Riemann-Hilbert decomposition. The notations are the same as in part I of this work (see the article reviewed above). The \(\ell\)-periodic reduction is a subfamily of the TL hierarchy with the additional constraints: \(L^{\ell}=\Lambda^{\ell},\) \(M^{- \ell}=\Lambda^{-\ell}.\) It is shown that the solutions of \((TL)_{\ell}\) hierarchy are independent of \(x_ n\), \(y_ n\) (n\(\equiv 0 mod \ell)\) and some equivalent forms for the \(\ell\)-periodic condition are given. The linearization scheme is also considered. Finally, the Riemann-Hilbert decomposition is considered. The wave matrices are written in the form \(W^{(0,\infty)}(x,y)=\hat V^{(0,\infty)}(x,y)\exp [\xi (x,\Lambda)+\xi (y,\Lambda^{-1})].\) Then \(\hat V^{(0)}=\hat V^{(\infty)}H(x,y),\) i.e. H(x,y) is decomposed to the upper triangular matrix \(\hat V^{(\infty)}\) and the lower triangular matrix \(\hat V^{(\infty)}\). It is claimed that the TL hierarchy of orthogonal or symplectic type and the multi-component TL hierarchy can also be considered.
scientific article; zbMATH DE number 3879430
scientific article