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Solitons and infinite dimensional Lie algebras. (English) Zbl 0557.35091
Since 1981 the authors have published in collaboration with Date and Kashiwara a series of papers devoted to the analysis of the algebraic structure of soliton equations. These and the works of Drinfeld, Sokolov, Wilson and others constitute a recent surge on the study of soliton theory. Starting from a Clifford algebra A the authors give the definitions on the Fock representation of A, Lie algebras \(A_{\infty},B_{\infty},C_{\infty}\) and \(D_{\infty}\), which are respectively the infinite dimensional analogues of the classical Lie algebras \(A_ n\), \(B_ n\), \(C_ n\) and \(D_ n\). They discuss their reductions to the Kac-Moody Lie algebras \(A_ l^{(1)},C_ l^{(1)},D_ l^{(1)},A_ l^{(2)}\) and \(D^{(2)}_{l+1}\). The authors construct the so-called \(\tau\)-function in each case basing on the multiplicative group in the Clifford algebra and its Fock representation. These \(\tau\)-functions are shown to satisfy bilinear identities which can be rewritten into a series of nonlinear differential equations for \(\tau\) (x). By introducing further transformations of dependent variables these equations turn out to be just soliton equations. The well-known soliton equations, such as the KP, KdV, MKdV, sine-Gordon, nonlinear Schrödinger, Heisenberg ferromagnet equations, the principal chiral model and many others including some difference equations can be generated in this way. This article contains also a number of useful tables such as the Chevalley basis for the above- mentioned Kac-Moody Lie algebras, a list of bilinear differential equations of low degree for the KP and modified KP hierarchies.
Reviewer: G.Tu

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
17B65 Infinite-dimensional Lie (super)algebras
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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