Flaschka, H.; Newell, A. C.; Ratiu, T. Kac-Moody Lie algebras and soliton equations. II: Lax equations associated with \(A_ 1^{(1)}\). (English) Zbl 0643.35098 Physica D 9, 300-323 (1983). The paper is concerned with the role of Lie algebras in soliton theory. The authors concentrate on the soliton equations associated with the AKNS eigenvalue problem \[ V_ x(x,\zeta)\quad = \quad \left[\begin{pmatrix} -i&0\\ 0&i\end{pmatrix} \zeta + \begin{pmatrix} 0&q(x)\\ r(x)&0 \end{pmatrix} \right]V\quad (x,\zeta)=^{def}PV \] and on their relationship with the loop algebra of formal series \(\Sigma X_ k\zeta^{-k}\), \(X_ k\in sl(2).\) Let \(V_ t=Q^{(N)}V\) be a time evolution for V, where \(Q^{(N)}=Q_ 0\zeta\) \(N+Q_ 1\zeta^{N-1}+...+Q_ N\) is a polynomial in \(\zeta\), \(Q_ s\in sl(2)\), \(Q=\sum^{\infty}_{j=0}Q_ j\zeta^{- j}=\sum^{\infty}_{j=0}\begin{pmatrix} h_ j & e_ j \\ f_ j & -h_ j\end{pmatrix} \zeta^{-j}\) is a formal series. All the \(Q_ j\) can be determined from the equation \(Q_ x=[P,Q].\) Main result 1: Equations \(Q_{t_ k}=[Q^{(k)},Q]\), \(Q^{(k)}=\sum^{k}_{j=0}Q_ j\zeta^{k-j}\) are commuting Hamiltonian flows on the Lie algebra \(\{\sum^{\infty}_{0}X_ j\zeta^{-j}:\) \(X_ j\in sl(2,{\mathbb{C}})\}\), with respect to a natural Lie-Poisson bracket. The next theorem shows that soliton equations in the generalized sense still have conservation laws. Main result 2: There exist polynomials \(F_{k_ j}\) in the \(e_ i\), \(f_ i\), \(h_ i\), such that \(\partial F_{k_ j}/\partial t_ e=\ni F_{e_ k}/\partial t_ m.\) [For part III see the following review.] Reviewer: I.Chekalov Cited in 2 ReviewsCited in 50 Documents MSC: 35Q99 Partial differential equations of mathematical physics and other areas of application 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Keywords:Kac-Moody algebras; loop algebra; Hamiltonian flows; soliton equations Citations:Zbl 0643.35099 PDFBibTeX XMLCite \textit{H. Flaschka} et al., Physica D 9, 300--323 (1983; Zbl 0643.35098) Full Text: DOI References: [1] Drinfel’d, V. G.; Sokolov, V. V., Dokl. Akad. Nauk SSSR, 258, 1, 11 (1981) [2] Wilson, G., Ergodic Theory and Dynam. System, 1, 361 (1981) [3] Kupershmidt, B. A.; Wilson, G., Invent. Math., 62, 403 (1981) [4] Hirota, R., Phys. Rev. Letters, 27, 1192 (1971) [5] RIMS preprints, 360, 362 (1981) [6] Segal, G.; Wilson, G., Loop groups and equations of KdV type (1983), preprint [7] J. Palmer and D. Pickerell, in preparation.; J. Palmer and D. Pickerell, in preparation. [8] Deift, P.; Trubowitz, E., Comm. Pure Appl. Math., 32, 121 (1979) [9] Dhooghe, P. F., Bäcklund equations on Kac-Moody-Lie algebras and integrable systems (1982), preprint · Zbl 0579.58012 [10] Adler, M.; van Moerbeke, P., Adv. Math., 38, 267 (1980) [11] Adler, M., Invent. Math., 50, 219 (1979) [12] Kostant, B., Adv. Math., 34, 195 (1979) [13] Symes, W. W., Invent. Math., 59, 13 (1980) [14] Ratiu, T., Springer Lecture Notes in Mathematics V, 219 (1980) [15] Gel’fand, I. M.; Dikii, L. A., Funkts. Anal. Prilozh., 10, 1, 26 (1976) [16] Gel’fand, I. M.; Dikii, L. A., Funkts. Anal. Prilozh., 13, 1, 8 (1979) [17] Gel’fand, I. M.; Dikii, L. A., Uspekhi Mat. Nauk, 30, 5, 67 (1975) [18] Gel’fand, I. M.; Manin, Yu. I.; Shubin, M. A., Funkts. Anal. Prilozh, 10, 4, 30 (1976) [19] Kaup, D. J.; Newell, A. C., J. Math. Phys., 19, 798 (1978) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.